{"meta":{"snapshot_schema":"p42-erdos-frontier-atlas-v1","atlas_version":"0.2.0","generated":"2026-07-13","source":"research_sessions/res_20260711_erdos_machinery_audit (cultural-soliton-observatory), 51 deep audits over 95 triaged Erdos prize problems","total":51,"survey":{"triaged":95,"deep_audited":51},"queue":{"ready_interfaces":13,"p42_packages":7,"recommended_unpacked":0,"reserve_unpacked":6},"facets":{"boardability":{"READY":13,"HEAVY":14,"NONE":24},"reach":{"MOVABLE":8,"UNKNOWN":2,"WALL":41},"lane":{"SAT+DRAT-nonexistence":6,"exact-backtracking":10,"witness-local-search":11,"LP/SDP-certificate":1,"wall":23},"p42":{"true":7,"false":44}},"board_class_rule":{"READY":"R1: exact integer/rational witness check, a-priori bounded on a byte-capped witness, <= ~1 s per candidate. R2: a concrete OPEN numeric frontier that a single submitted finite witness strictly improves. R3: witness conditions discrete or open/robust (strict inequalities) so integer/rational witnesses are lossless. R4: one board per finite frontier object (duplicates cross-reference the carrier).","HEAVY":"Exact adjudication exists but fails R1 timing (long MIS/SAT run per candidate) or the movable claim is certificate-shaped (DRAT chain, exhaustion receipt, SDP dual) rather than a small witness. Optimistic-oracle tier.","NONE":"No exact poly-time verifier in any movable direction, or witness not finitely representable, or no open finite frontier (asymptotic-only / solved / duplicate object). Do not board; usually also a do-not-enter wall (see atlas/walls.md)."},"provenance":{"repository":"https://github.com/techno-optimist/erdos-frontier-atlas","commit":"49e70f25e4bab48969293059cb9c95674de2aec0","path":"atlas/problems.json","sha256":"ed1148db84bf33d9d3beb0f5cba24e9a84dfd30314527fc800a57bc291d08e80","license":"MIT","copyright":"Copyright (c) 2026 Kevin Russell"},"settlement_authority":false},"entries":[{"id":1,"title":"Erdős distinct subset sums (min largest element, A276661)","prize":"$500 (for the asymptotic headline conjecture only; the finite a(10) determination carries no prize)","statement":"Erdős #1 headline: if A⊆{1,…,N}, |A|=n, has all 2^n subset sums distinct, then N ≫ 2^n (asymptotic, $500). The IN-CLASS finite sub-problem (OEIS A276661): a(n) = the minimal possible largest element of a size-n set with distinct subset sums.","finite_object":"A size-11 set of positive integers with all 2048 subset sums distinct, scored by its largest element. Improve the a(11) upper bound with a smaller witness, or raise the lower bound with a reproducible exhaustive-search certificate.","current_record":"Exact terms a(0..10) = 0,1,2,4,7,13,24,44,84,161,309. Paul W. Dyson proved a(10)=309 exactly in 2025 (OEIS A276661). The live finite frontier is a(11): 310 <= a(11) <= 594. The lower bound follows by deleting the largest element from any 11-set and using a(10)=309; the upper bound is the Conway-Guy 11-set.","beatable":"MOVABLE","beatable_reason":"The live a(11) frontier is finite and two-sided. Witness search can lower 594 immediately; exact backtracking can raise the 310 lower bound or close the term. The constraint has no known compact CNF/DRAT encoding, so lower-bound claims require complete, reproducible search transcripts.","fit_score":6,"impact_score":4,"impact_reason":"Erdős #1 is famous, while the movable a(11) term is a non-prize finite record problem. A better witness or certified bound would be a citable OEIS-table advance.","verifier":"Given a candidate 11-set, enumerate all 2048 subset sums and test pairwise distinctness with exact integers. Witness verification is deterministic and fast; optimality remains the expensive search claim.","attack":"Run dedicated parallel backtracking and local search for an 11-set with maximum below 594. Maintain subset sums incrementally, prune on collision, and split workers by canonical prefixes. For lower bounds, publish complete shard definitions, deterministic transcripts, and an independent checker.","our_edge":"Verifier-first witness search and sharded exhaustive search fit directly. The witness checker is tiny; the value lies in auditable search orchestration and reusable negative-result receipts. No short DRAT-style certificate is currently known for the lower-bound side.","verdict":"PARTIAL. The $500 asymptotic headline is not resolved by this finite board. The live in-class target is the open exact term a(11) in [310,594]; improvements are legitimate record contributions but do not claim the headline prize.","lane":"exact-backtracking","board_class":"READY","board_class_reason":"witness = 11 integers; 2048 subset sums, exact and fast; live bracket 310 <= a(11) <= 594","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/1","links":{"oeis":["A005318","A276661"],"arxiv":["2208.12182","2308.03748"]},"frontier":{"summary":"310 <= a(11) <= 594","value":"310 <= a(11) <= 594","holder":"Conway-Guy set (upper); Dyson a(10)=309 + drop-largest lemma (lower)","year":2025,"note":"a(10)=309 PROVEN EXACT by Paul W. Dyson (OEIS A276661, Oct 21 2025) — the n=10 target is CLOSED; board and attack target moved to a(11)."},"p42_slug":"distinct-subset-sums-a11"},{"id":2,"title":"Covering-system min-modulus record (>42)","prize":"$1000","statement":"Erdős's \"favourite problem\": can the smallest modulus of a covering system (distinct moduli covering all integers) be arbitrarily large? DISPROVED (Hough 2015: ≤10^16; BBMST 2022: ≤616000). Remaining finite sub-problem: the lower-bound record — a covering system whose smallest modulus exceeds the current best of 42.","finite_object":"A covering system (finite set of congruences a_i mod m_i) with distinct moduli all ≥ 43 that covers every integer. \"Finite\" only nominally: Nielsen's min-40 witness already has >10^50 congruences; a min-43+ witness is comparably or vastly more astronomical.","current_record":"Smallest modulus = 42, Tyler Owens, MSc thesis, BYU, 2014 (improving Pace Nielsen's 40, 2009; Krukenberg/Choi/Gibson chain before that). Upper bound side: Hough 2015 (≤10^16), Balister–Bollobás–Morris–Sahasrabudhe–Tiba 2022 (≤616000), Cummings–Filaseta–Trifonov (≤118 for squarefree moduli).","beatable":"WALL","beatable_reason":"The witness is not an explicitly-enumerable finite list: Nielsen's min-modulus-40 system already exceeds 10^50 congruences, defined recursively (Filaseta-school hierarchy: admit multiples of prime p only after exhausting moduli with all prime factors 10^50 for the WEAKER 40 record), exceeding any solver's representable instance size by dozens of orders of magnitude. A structured recursive construction (extending Nielsen/Owens) is a pure number-theory research program, not a verifier-first CPU-farm search — no amount of ~100-core compute or DRAT certification bears on it. Rough compute to beat 42: not a compute question; it is an open construction-theory problem.","our_edge":"none beyond generic compute — and generic compute is inapplicable here since the object cannot be represented for search.","verdict":"PARTIAL — headline conjecture is asymptotic and already DISPROVED (not finite-computable), but a genuine finite sub-problem exists (push the min-modulus lower-bound record above 42). That sub-problem is a hard WALL for our verifier-first search farm: the witness has >10^50 recursively-defined congruences and is not representable in any SAT/ILP/MIS/exhaustion instance, the verifier is not clean poly-time residue enumeration, and the record was set by bespoke number-theoretic recursion we hold no edge in. Do not attack.","lane":"wall","board_class":"NONE","board_class_reason":"witness not representable: a min-modulus-40+ covering system already has > 10^50 recursively-defined congruences; no flat poly-time check","wall_reason":"witness not representable: a min-modulus-40+ covering system already has > 10^50 recursively-defined congruences; no flat poly-time check","erdos_url":"https://www.erdosproblems.com/2","links":{"oeis":[],"arxiv":[]}},{"id":13,"title":"Erdős–Sárközy no-term-divides-sum sets (|A|≤N/3+O(1))","prize":"$100 (already awarded to Bedert)","statement":"A⊆{1..N} with no a,b,c∈A such that a|(b+c) and an colors; the extremal object is the complete set of χ=n linear spaces or the smallest m_k config with χ≥n+k.","current_record":"Asymptotic headline PROVED for all sufficiently large n: Kang, Kelly, Kühn, Methuku, Osthus (FOCS 2021; Annals of Math 198(2):537–618, 2023); prior Kahn (1992) (1+o(1))n. Small-case computational frontier: fully verified n≤12 — Hindman n≤10 (1981), Romero–Alonso-Pecina n≤12 (2014, heuristic, no proof), Kirchweger–Peitl–Szeider (SAT 2023, first DRAT-certified) full n≤12 plus only low-m/high-m bands for n=13–18 (middle-m unsolved at 3 CPU-days/instance; hardest solved single instance n=15,m=85 ≈2.6 CPU-days). χ=n extremal linear spaces completely classified for n≤12 (KPS Thm 15: H_{n,k} chain + Fano + sporadics).","beatable":"WALL","beatable_reason":"Three independent reasons the meaningful frontier does not move for us. (1) The conjecture is TRUE — proved for all large n (KKKMO, Annals 2023) and verified for all n≤12 — so the witness branch (find a χ>n hypergraph) is a KNOWN-EMPTY dead end; there is no record-beating finite object to find, unlike kissing/R(5,5)-lower-bound where a witness might exist. (2) The current small-case record is held by EXACTLY our toolchain (Kirchweger–Peitl–Szeider 2023: SAT + SAT-Modulo-Symmetries + co-certificate-learning + DRAT), the same R(5,5)-was-already-at-the-wall situation. (3) The certification/UNSAT branch hits a real complexity wall, not a budget wall: KPS could not even get a SAT solver to prove K₁₃ is not 12-edge-colorable, because that reduces to the pigeonhole formula, which is exponentially hard for resolution/CDCL — a 100-core farm does nothing against pigeonhole. Only a narrow, low-value sliver is technically movable: closing n=13's unsolved middle-m band to full verification, a non-prize incremental note.","fit_score":5,"impact_score":3,"impact_reason":"The problem is famous (one of Erdős's favorite three, $500, culminating in a 2023 Annals result), but the finite-computation contribution actually available to us is low: an incremental one-step small-case extension (n=12→13) using a published pipeline, non-prize (the prize is settled asymptotically), of minor field significance and no real-world relevance. The high-value mathematical content (asymptotic proof, extremal classification n≤12) is already done.","verifier":"Verifying a PROPOSED n-edge-coloring is trivial poly-time: build the intersection (line) graph and check every intersecting pair got distinct colors, O(m²). BUT the load-bearing direction — proving a bucket (n,m) has NO non-colorable member, i.e. non-colorability of the hard instances — is coNP and empirically pigeonhole-hard (CDCL exponential). KPS pipeline emits a DRAT certificate for the whole (n,m) exhaustion, independently checkable.","attack":"Run the open-source KPS SMS+CCL pipeline on ~100 cores to close the gray middle-m band at n=13 (and possibly low bands of n=14) with DRAT certs. Rough compute: hardest solved instance was ~2.6 CPU-days; the unsolved middle band is many buckets each plausibly 10–100× harder → farm-months, and the K₁₃-not-12-colorable-style coloring-UNSAT buckets are pigeonhole-hard and likely infeasible at ANY realistic core count. Net: a possible incremental \"EFL verified for n≤13\" note, no prize, no theorem, blocked from going meaningfully further by the resolution wall.","our_edge":"None beyond generic compute plus the already-public KPS SMS/DRAT pipeline. The record sits on exactly our verifier-first SAT+DRAT stack, and the conjecture's TRUE status removes the witness target entirely.","verdict":"PARTIAL — headline conjecture is asymptotically PROVED (new-theorem territory, NONE for us), but a genuine in-class finite sub-problem exists (SAT+DRAT small-case verification and χ=n extremal enumeration) that maps perfectly to our farm mechanically. The catch: the record is already set by our own tools, the witness branch is known-empty because the conjecture is true, and the certification branch is pigeonhole-walled — so the only movable target is a marginal, non-prize n=13 note. Mechanics fit great; payoff does not. Do not launch a campaign.","lane":"SAT+DRAT-nonexistence","board_class":"HEAVY","board_class_reason":"the movable n=13 step is a whole-bucket DRAT certificate (pigeonhole-hard, resolution-exponential); coloring witnesses certify nothing new","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/19","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"EFL fully verified n <= 12; the n=13 bucket is a whole-family DRAT-certificate claim (pigeonhole-hard CDCL)"}},{"id":20,"title":"Erdős–Rado sunflower conjecture (f(n,k) < c_k^n)","prize":"$1000 (headline asymptotic conjecture only; finite table-extension carries no prize)","statement":"Let f(n,k) be minimal such that every n-uniform family with |F| ≥ f(n,k) contains a k-sunflower (k sets with all pairwise intersections equal). Conjecture: f(n,k) < c_k^n for some c_k depending only on k. Headline is asymptotic ($1000, OPEN, page-marked non-finite). Genuine finite sub-problem: the exact \"sunflower numbers\" Sun(m,n) (OEIS A332077, tagged hard/more) — smallest number of ≤m-sets forcing an n-sunflower — equivalently the largest s-sunflower-free m-uniform family.","finite_object":"A family of N sets, each of size ≤ m, on a fixed ground set [g], containing no s-set that forms a sunflower (equal pairwise intersections). Target: beat a recorded best-known size for a specific (m,s) cell of the A332077 table, or improve the base gadget behind the sqrt(10)^k growth-constant lower bound.","current_record":"Sun(3,3)=21 PROVEN (Erdős–Rado; extremal 20-set family). Open lower bounds only for larger cells: Sun(4,3)≥55 (Abbott–Hanson–Sauer 1972, 54-set 4-uniform construction), Sun(3,4)≥39 (Abbott–Exoo 1992, 38 triples), Sun(3,5)≥89 (Abbott–Hanson 1974, 88 triples). Row m=2 fully solved (AHS closed form). Growth-rate lower bound lim f(k)^{1/k} ≳ sqrt(10)≈3.162 via AHS product recursion. No 2023–2026 improvement to any of these lower bounds found; recent activity is all on the asymptotic upper side (ALWZ / Bell–Chueluecha–Warnke (Cs·log k)^k).","beatable":"MOVABLE","beatable_reason":"Only the LOWER-BOUND (witness) direction is in reach; the exact value / upper bound is a WALL (ground set is unbounded, so certifying no larger family exists needs a ground-set-size theorem, not a fixed-vertex SAT/DRAT closure — the R(5,5) asymmetry). The witness side is genuinely pushable: the recorded lower bounds (54, 38, 88 sets) are 30–50-year-old constructions, and Abbott–Exoo's 1992 record was set on 1992-era computer search — a multi-generation solver/hardware gap we could exploit. But these are structured, algebraic, Exoo-class records (product/recursive constructions), not soft empty-board numbers, so they are sticky; a real chance exists on the smaller cells (Sun(3,4), Sun(3,5), Sun(4,3)) but no guarantee of beating highly-tuned algebraic constructions with search.","fit_score":5,"impact_score":4,"impact_reason":"Headline is a celebrated $1000 Erdős problem in a hot area (post-ALWZ sunflower breakthrough), but it is untouchable by finite computation. Our only achievable deliverable is extending an OEIS \"hard/more\" table by a lower-bound term or nudging the sqrt(10) growth constant — a real, citable but minor contribution: no prize, modest field significance, no real-world relevance.","verifier":"Given N sets: certify no s of them form a sunflower. For s=3, enumerate triples grouped by common core — O(N^3 · m) worst case, far less with core-bucketing. Exact, poly-time for fixed s, cheap (seconds for N in the hundreds).","attack":"Per (m,s) cell, fix ground set [g] and sweep g upward. Build the s-uniform conflict hypergraph on nodes = all C(g,m) m-subsets, hyperedge per sunflower s-tuple (enumerate by fixing a core C0 and picking sets ⊇ C0 with pairwise-disjoint petals). Solve max set-packing-with-ternary-conflicts via (a) simulated annealing / local search seeded from the known record construction to hunt for a +1 extension — hours on a few cores, the natural first strike; and (b) ILP (Gurobi) or MaxSAT with S_g symmetry-breaking for a fixed-g optimum. Sizes: C(24,4)=10626 vars, millions of sunflower-triple constraints — heavy but feasible on ~100 cores over days per cell. Deliverable: an improved witness → new A332077 lower-bound term. Rough compute: SA seeding hours; full ILP/MaxSAT sweep across 3–4 cells and several g values ≈ days–weeks on 100 cores.","our_edge":"None beyond generic compute plus a ~30-year solver/hardware-generation gap over the last (Abbott–Exoo 1992) computer search. Our verifier-first harness applies cleanly, but the extremal objects here are algebraic/structured, which is historically where human construction beats blind search — so no domain edge.","verdict":"PARTIAL. The $1000 headline (f(n,k)= 55 (1972), Sun(3,4) >= 39 (1992), Sun(3,5) >= 89 (1974) - construction records static 30-50 yr"}},{"id":21,"title":"q(6): min edges, 6-uniform intersecting hypergraph with τ=6","prize":"$500 — but already PAID to Kahn (1994) for the asymptotic linear upper bound. Determining q(6) claims NO prize; it is a real but non-prize combinatorial contribution.","statement":"f(n)=q(n) is the minimum number of edges in an n-uniform intersecting hypergraph H (every pair of edges meets) with maximal covering number τ(H)=n (no n−1 vertices meet every edge). The headline Erdős–Lovász question — is f(n) ≪ n? — was PROVED affirmatively by Kahn (1994), claiming Erdős's $500. The live FINITE frontier is the exact small-value table: f(1)=1, f(2)=3, f(3)=6, f(4)=9, f(5)=13, and f(6) is OPEN, bracketed 14 ≤ f(6) ≤ 18.","finite_object":"A 6-uniform intersecting hypergraph with m edges (14 ≤ m ≤ 18) and covering number exactly 6 — i.e. pairwise-intersecting 6-sets such that no 5 vertices hit all edges. For m=14, Barát's structural bound gives at most 29 vertices. q(6) is the least such m.","current_record":"q(3)=6, q(4)=9 (Tripathi 2014); q(5)=13 (Barát–Wanless, J. Combin. Designs 2021, arXiv:2011.04444). q(6) is undetermined with 14 ≤ q(6) ≤ 18: Sivashankar's 2026 bound 3r−4 gives the lower endpoint 14, while the Barát–Wanless PG(2,5) construction gives 18. The prior exhaustive search was storage-limited on 2014 consumer hardware. Our July 2026 SAT/DRAT attempt failed on the global transversal constraint; the live route is orderly generation with canonical augmentation and tau-deficiency pruning.","beatable":"MOVABLE","beatable_reason":"The bracket is narrow: 14 ≤ q(6) ≤ 18, so any valid witness with 14–17 edges improves the record and a 14-edge witness settles the value. Barát's exhaustive campaign stopped at a storage wall on 2014 consumer hardware. Our July 2026 audit falsified the tempting SAT/DRAT route: the global tau constraint caused D(5,12) and every bounded D(6,14) encoding to time out or explode. The remaining computational route is orderly generation with canonical augmentation, degree/vertex bounds, and tau-deficiency pruning, paired with the cheap exact witness verifier. Exhaustive nonexistence remains potentially hard and requires auditable shard coverage rather than an assumed compact DRAT certificate.","fit_score":8,"impact_score":4,"impact_reason":"Prize weight ≈0 (the $500 is spent on the asymptotic, which is settled). Field significance is modest and niche: extends a known extremal table (q(3..5) settled; q(6) the next open entry, with q(7) a stretch) in extremal set / design theory — the kind of exact-value result that gets cited within that subfield and updates the Erdős-problems page, but is not a paradigm mover. Real-world relevance: none. Value is a clean, certified, publishable small-case resolution, not a headline.","verifier":"Poly-time exact. Given edge list (m 6-sets): (a) pairwise-intersecting — O(m²·6); (b) τ=6 i.e. no 5-cover — a fixed-size-5 hitting-set check over ≤N vertices (N≲6m), either C(N,5) enumeration or a tiny ILP; unsatisfiable ⇒ τ≥6, and τ≤6 is automatic since any edge covers. Cost: milliseconds per candidate.","attack":"Orderly generation for m=14,15,16,17 using canonical augmentation of 6-edges up to isomorphism. Apply Barát–Wanless degree and vertex bounds (for m=14, V≤29), reject branches with disjoint edge pairs, and maintain a monotone tau-deficiency bound that prunes partial families already forced to admit a 5-cover. Every surviving leaf is checked by the independent exact pairwise-intersection/no-5-cover verifier. Shard the canonical search space with deterministic manifests and replayable coverage receipts; a witness with m≤17 improves the upper bound, while an exact value requires complete audited exhaustion below the first witness. Do not reuse the failed global SAT/DRAT encoding.","our_edge":"A verifier-first orderly-generation harness can combine nauty-style canonical augmentation, exact 5-cover rejection, Barát's structural bounds, deterministic sharding, and replayable coverage receipts on hardware far beyond the storage-limited 2014 campaign. The edge is engineering and auditability, not the already-falsified R(5,5) SAT analogy.","verdict":"STRONG but route-specific — q(6) ∈ [14,18] is a finite movable number with a cheap exact witness verifier. SAT/DRAT is a recorded wall for our encoding; only orderly generation/canonical augmentation with tau-deficiency pruning is authorized for further compute. A 14–17 edge witness is directly publishable progress, while exact nonexistence claims require complete verified-search coverage.","lane":"exact-backtracking","board_class":"READY","board_class_reason":"witness = <= 64 six-sets; pairwise-intersect + complete 5-cover branch-and-bound <= 9,331 nodes, ms-exact; open bracket 14..18 witness-improvable (m <= 17 wins)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/21","links":{"oeis":[],"arxiv":["2011.04444","2503.14918","2606.24878"]},"campaign_finding":"2026-07-12: we attempted q(6) with our SAT+DRAT pipeline (the R(5,5) method). It FAILED: q(r) is tau-critical, valid objects are vanishingly rare among m-edge intersecting families, and the tau>=r encoding is a C(V,r-1) blowup whose resolution proofs explode. D(5,12) (proving the KNOWN q(5)>=13) timed out at 3600s/6.4GB; D(6,14) timed out at every proven vertex count V in [21,29] (up to 8.4M clauses) — kissat could not even decide it. R(5,5) worked because its constraints are LOCAL (SAT-friendly); q(r)'s are GLOBAL/rare (SAT-hostile). The correct tool is ORDERLY GENERATION (canonical augmentation + tau-deficiency prune), which is what Barat used. The P42 board STAYS (the exact verifier works fine) as a MARKET-MAKER board: the frontier is movable, but by generation or cluster cube-and-conquer, not by our SAT machinery. Vertex bounds for the generation attack: Barat Lemma 6.4 (e=14 => V<=29), Obs 6.1/6.2 degree caps; Sivashankar q(6)>=14 means a single 14-edge witness would give q(6)=14 exactly.","frontier":{"summary":"14 <= q(6) <= 18","value":"14 <= q(6) <= 18","holder":"Sivashankar 3r-4 (lower, arXiv:2606.24878 Jun 2026); Barát m(6)=18 unique PG(2,5) construction (upper, arXiv:2011.04444)","year":2026},"p42_slug":"q6-intersecting-hypergraph"},{"id":27,"title":"ε-almost covering systems with moduli in [N, CN]","prize":"$100","statement":"Is there a constant C>1 such that for every ε>0 and N≥1 there is an ε-almost covering system (congruences a_i mod n_i, distinct moduli, density of uncovered integers ≤ ε) with all moduli in [N, CN]? Headline answer: NO.","finite_object":"A covering system (or ε-almost covering system) of Z with distinct moduli, congruences a_i mod n_i, with all moduli ≥ N; the \"record\" instance is the largest achievable minimum modulus.","current_record":"Headline #27: DISPROVED (Filaseta, Ford, Konyagin, Pomerance, Yu, 2007). Related finite frontier — largest known minimum modulus of a distinct covering system = 42 (D. Owens, 2014 BYU MS thesis, structured construction), improving Nielsen's 40 (2009); Krukenberg/Gibson before. Upper bound on any covering's min modulus: ≤ 616000 (Balister, Bollobás, Morris, Sahasrabudhe, Tiba, 2022, \"distortion method\"), improving Hough's 10^16 (2015). The 42→616000 gap is closed by neither construction nor search today.","beatable":"WALL","beatable_reason":"The headline (a uniform constant C for all ε,N) is DISPROVED by Filaseta–Ford–Konyagin–Pomerance–Yu (2007) via an asymptotic density theorem — pure analysis, no finite witness, nothing for a farm to push. The only genuinely finite sub-problem in-class is the covering-system minimum-modulus construction record, and that is itself a wall for our specific machinery: (a) the record was set by structured algebraic (prime-power tower / Krukenberg–Gibson distributed-covering) constructions plus human insight, not flat combinatorial search; (b) the witness is not a tractable object — a covering system with minimum modulus 42 already has well over 10^50 distinct moduli and an LCM period that is astronomically large, so there is NO flat poly-time \"enumerate a period and check every residue is hit\" verifier (verification is done by recursive distributed-covering structure, not enumeration); (c) you cannot even encode the search as a SAT/ILP instance — that would need >10^50 variables. This is the R(5,5)-upper-bound situation, not the R(5,5)-witness situation: the frontier is bounded by a theorem (min modulus ≤ 616000, BBMST 2022) but the practical construction ceiling is a combinatorial-explosion wall, not a compute wall we can climb.","fit_score":2,"impact_score":3,"impact_reason":"$100 prize is already claimed (disproved 2007). Field significance of the residual finite frontier (min-modulus record 42) is a niche number-theory curiosity; nudging it to 43 would be a cute non-prize milestone with little downstream or real-world relevance, and the deep questions (true threshold in [42, 616000]) are theorem-shaped, not witness-shaped.","verifier":"Checking that a proposed system covers Z (or leaves density ≤ ε). But the natural extremal object has >10^50 moduli and an astronomically large LCM period, so there is no flat enumerable poly-time checker over a period; verification requires the recursive distributed-covering structure of the construction itself. This absence of a clean small-object exact verifier is the disqualifier for our verifier-first farm.","attack":"No viable attack for our tools. A flat SAT/ILP/MIS encoding is impossible (witness has >10^50 moduli). Structured local-search over Krukenberg–Gibson building blocks to nudge the min-modulus record 42→43 is conceivable but is exactly the algebraic-construction game where humans have been stuck since 2014, not a CPU-farm-shaped problem, and success is not certifiable by a cheap exact verifier. A tractable but non-record-bearing exercise would be: exhaustively compute, for tiny N and modest C, the minimal C(N) admitting an ε-almost cover with moduli in [N,CN] (ILP over LCM(N..CN) for small N where the period is enumerable) — genuine PARTIAL-exploratory work, but no established record to beat and no prize.","our_edge":"None beyond generic compute. This class rewards algebraic/analytic construction (distortion method, prime-power towers), which is orthogonal to our verifier-first witness-search precedents (kissing, R(5,5), LP/SDP bounds). We have no exact small-object verifier here, which is the one asset our farm is built around.","verdict":"WALL / NONE for our machinery. The $100 headline is an already-disproved asymptotic theorem (no finite witness). The one in-class finite frontier — the covering-system minimum-modulus construction record (42, Owens 2014) — fails our fit test on every axis: no tractable exact verifier (witness >10^50 moduli, LCM period astronomically large), no SAT/ILP-encodable search space, and a record set by structured algebraic construction rather than flat search. Analogous to the R(5,5) UPPER bound (NONE), not the R(5,5) witness (STRONG). Recommend PASS. The only honest computational contribution available (small-N exact ε-almost-cover thresholds) is exploratory, record-free, and low-value.","lane":"wall","board_class":"NONE","board_class_reason":"headline DISPROVED (Filaseta-Ford-Konyagin-Pomerance-Yu 2007); extremal witnesses not finitely representable","wall_reason":"headline DISPROVED (Filaseta-Ford-Konyagin-Pomerance-Yu 2007); extremal witnesses not finitely representable","erdos_url":"https://www.erdosproblems.com/27","links":{"oeis":[],"arxiv":[]}},{"id":30,"title":"Erdős #30 — max Sidon set h(N)","prize":"$1000","statement":"h(N) = max size of a Sidon set (B2 set, all pairwise sums distinct) in {1,...,N}. Conjecture: for every ε>0, h(N) = N^{1/2} + O_ε(N^ε).","finite_object":"Two candidates: (A) an exact extremal Sidon set / optimal Golomb ruler of a given number of marks (A003022/A005282) — trivial O(k^2) verifier for constructions; (B) a rational certificate for an upper-bound constant b in h(N) ≤ √N + b·N^{1/4} + O(1) (finite convex-QP dual over smoothing kernels, checkable in exact arithmetic).","current_record":"EXACT VALUE: optimal Golomb rulers / max Sidon sets proven optimal up to 28 marks — OGR-28 (length 585) proven by distributed.net exhaustive branch-and-bound, completed Nov 23, 2022 after 8.5 years on a global volunteer grid; OGR-27 in 2014. UPPER-BOUND CONSTANT: h(N) ≤ √N + 0.9435·N^{1/4} + O(1), \"Vector-valued smoothing for finite Sidon sets\" (arXiv 2607.01169, July 2026, eight-kernel rational certificate); prior 0.97633 (Carter, Georgiev, Gómez-Serrano, Hunter, O'Bryant, Tao, Wagner with AlphaEvolve, 2025) and 0.98183 (Carter–Hunter–O'Bryant, Acta Math. Hungar. 2025). LOWER BOUND: √N(1−o(1)) (Singer / Erdős–Turán, classical).","beatable":"WALL","beatable_reason":"see above","fit_score":5,"impact_score":6,"impact_reason":"Famous Erdős problem, deep in additive combinatorics, actively worked by Terence Tao; the exact-value side (Golomb rulers) has real-world use (radio-astronomy antenna arrays, error-correcting codes, sensor placement). Prize is only $1000 and the headline is unresolvable by computation. Field significance is real, but the contribution WE could make (a decimal on a constant already being pushed by a stronger consortium) is negligible.","verifier":"(A) Construction check: enumerate all C(k,2) pairwise differences of a claimed size-k Sidon set, confirm distinctness — O(k^2), trivial. But the RECORD is the proven-optimal value, whose nonexistence side has no compact certificate (distributed.net emits a brute-force search log, not a DRAT-style checkable proof). (B) Upper-bound constant: verify a finite rational certificate (dual feasibility of the kernel QP) in exact arithmetic — cheap, seconds.","attack":"If forced: (A) SAT/ILP or specialized branch-and-bound for \"Golomb ruler with 29 marks, length ≤ L\" — but this is the OGR-29 frontier, astronomically beyond a 100-core farm (OGR-28 took 8.5 yr on a global grid; SAT/ILP scales far worse than distributed.net's dedicated pruner). Not viable. (B) Reproduce/extend the kernel-QP: choose m cooperating smoothing kernels, solve the convex quadratic program for the boundary majorant, round the numerical optimum to a finite rational dual certificate, verify in exact arithmetic — this is exactly our autoconvolution/PNT-LP pipeline (~hours on a few cores). Mechanically feasible but yields at best a decimal below 0.9435, which the Tao/AlphaEvolve consortium is already driving down; no defensible lead and no prize.","our_edge":"None beyond generic compute. Frontier B maps cleanly to our LP/SDP rational-certificate farm (autoconvolution C1≤1.50285 precedent, and #30's bound is in the same autoconvolution constant family), but the same method is being run faster and further by Tao + DeepMind. Frontier A needs a global-scale exhaustive grid we do not have.","verdict":"PARTIAL problem, but NOT actionable for us. The $1000 conjecture is asymptotic → NONE. Two genuine finite sub-problems exist: (A) exact extremal values / Golomb-ruler table extension — trivial construction verifier but the record is a compute WALL (distributed.net OGR-28, 8.5 yr on a global grid, no checkable nonexistence cert; OGR-29 unreachable by a 100-core farm); (B) the upper-bound constant b in √N+b·N^{1/4} — a finite exact-certificate optimization squarely in our autoconvolution/LP wheelhouse, but a saturated, monthly-moving lane owned by Tao + DeepMind AlphaEvolve + O'Bryant (0.9435 as of July 2026) where we have zero edge. Recommend PASS: no movable frontier for our machinery.","lane":"exact-backtracking","board_class":"HEAVY","board_class_reason":"frontier is proven-optimal (OGR-28); the next move is an optimality/exhaustion claim adjudicated by receipts, not a witness","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/30","links":{"oeis":[],"arxiv":["2607.01169"]},"frontier":{"summary":"OGR-28 (length 585) proven optimal Nov 2022 by distributed.net after 8.5 yr; next move (OGR-29) is an exhaustion claim"}},{"id":39,"title":"Infinite near-maximal-density Sidon set","prize":"$500 (Erdős)","statement":"Is there an infinite Sidon (B_2) set A ⊂ ℕ with |A ∩ {1,...,N}| ≫_ε N^{1/2-ε} for all ε>0?","finite_object":"Densest finite Sidon set in [1,n] ≡ optimal Golomb ruler (OGR) / perfect-ruler table (OEIS A004137). The headline object itself (an infinite set of density N^{1/2-ε}) is NOT finite.","current_record":"HEADLINE (asymptotic, no finite witness): best construction Ruzsa 1998, |A∩[1,N]| ≫ N^{√2−1+o(1)}, exponent ≈0.4142 (probabilistic, via {log p : p prime}); still the record as of 2025 (no improvement 2023–2026); Erdős proved liminf |A∩[1,N]|/√N = 0 for every infinite Sidon set. FINITE in-class table: optimal Golomb rulers to OGR-27/OGR-28 via distributed.net multi-year exhaustive B&B (OGR-28 ≈2022); OEIS A004137 perfect rulers proven to a(26) by Schwartau, Schröder, Wolf & Schöbel (2021).","beatable":"WALL","beatable_reason":"Two independent walls. (1) The $500 headline is purely asymptotic: improving the density exponent from √2−1≈0.4142 toward 1/2 needs a new construction/theorem, and no finite object can resolve it (an infinite set can't be exhibited by finite compute, and finite maxima provably don't nest into a dense infinite set — Erdős's liminf=0 result). (2) The genuine in-class finite sub-problem (densest finite Sidon set / optimal Golomb ruler table) is already at the feasibility wall, set by EXACTLY the exhaustive branch-and-bound tools we would use, run by distributed.net at a scale orders of magnitude beyond a ~100-core farm over multiple years; the next term (OGR-29) is estimated at years of that-scale compute. And even pushing that table one mark would not advance the prize conjecture at all. This is R(5,5)-witness-like: computational-looking but already exhausted.","fit_score":6,"impact_score":3,"impact_reason":"Headline is a classic named Erdős problem prestigious in additive combinatorics but only $500 and untouchable by us. The in-class finite piece (OGR/A004137 extension) is a minor OEIS-table contribution with low field significance and no real-world relevance, and does not bear on the prize.","verifier":"Sidon/Golomb-ruler check: all C(k,2) pairwise differences distinct — O(k^2) exact integer arithmetic. Trivially poly. Optimality of a size/length claim requires exhaustive nonexistence over all shorter rulers (branch-and-bound), which is the expensive part.","attack":"If attempted, the only finite target is extending the OGR / max-Sidon table by one mark: model as exact max-independent-set / branch-and-bound exhaustion with symmetry breaking (canonical augmentation), producing an optimal ruler + a DRAT-style nonexistence certificate for all shorter lengths. Rough compute: the current frontier (OGR-28) already consumed distributed.net-scale (>>10^4 core-years equivalent); OGR-29 is out of reach for a 100-core farm by many orders of magnitude. Local search / simulated annealing can find good Sidon sets cheaply but cannot certify optimality, so yields no record. No feasible path on either half.","our_edge":"None beyond generic compute. The finite frontier was set by distributed.net running a larger-scale version of our own exact exhaustive-search tools; we have no algorithmic or hardware advantage over it, and the headline needs a theorem we have no lever on.","verdict":"PARTIAL — headline is asymptotic/analytic (WALL: needs a new density construction, no finite witness). A real in-class finite sub-problem exists (densest finite Sidon set / optimal Golomb ruler table, exact poly verifier, our exact-search machinery), but its record is already at the compute wall set by distributed.net-scale exhaustive search — unbeatable by a ~100-core farm — and extending it would not advance the $500 conjecture. Net: no attackable frontier for us.","lane":"exact-backtracking","board_class":"HEAVY","board_class_reason":"optimality claims (branch-and-bound nonexistence receipts) are the movable half; witness side has no open tracked value","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/39","links":{"oeis":["A004137"],"arxiv":[]},"frontier":{"summary":"Sidon/Golomb optimality beyond OGR-28: exhaustion-certificate tier; asymptotic headline (Ruzsa exponent ~0.4142) untouchable"}},{"id":41,"title":"Distinct triple-sum (B_3) sets — infinite density liminf","prize":"$500 (headline only; unreachable by finite computation — asymptotic theorem required)","statement":"For an infinite A⊆ℕ with all triple sums a+b+c distinct (bar trivial coincidences), is liminf |A∩{1..N}|/N^{1/3}=0? ($500). Headline is asymptotic/analytic. Finite in-class analogue: the shortest B_3 ruler for n marks (OEIS A227358) — exact values known only through a(10)=309; a(11) is OPEN.","finite_object":"An 11-mark B_3 \"Golomb-like\" ruler: a set 0=b_00? (The sum-product conjecture.)","finite_object":"A finite set A ⊂ ℤ_{>0} of size n minimizing the sum-product functional — either R(A)=|A+A ∪ AA| (OEIS A263996) or max(|A+A|,|AA|). The verifiable witness is the explicit n-element set.","current_record":"TWO distinct frontiers. (1) ASYMPTOTIC (the actual $250 problem): max(|A+A|,|AA|) ≫ |A|^{1270/951 − o(1)}, 1270/951 ≈ 1.3354, Bloom [Bl25] — a theorem, NOT a finite computation. (2) FINITE small-case table: OEIS A263996 = smallest |A+A ∪ AA| over n positive integers; b-file gives every term to n≈205 (a(7)=26 via {1,2,3,4,6,8,12}, etc.). Best-known constructions for n up to 1000 were produced by Al Zimmermann's Programming Contest \"Sums and Products\" (Nov 2015–Feb 2016, 1,639 entries), a distributed heuristic-optimization competition. No finite-side movement found 2023–2026.","beatable":"WALL","beatable_reason":"The headline is explicitly asymptotic and self-declares \"cannot be resolved with a finite computation\" — no witness touches the $250 exponent (a single better construction cannot change an ε-exponent claim; that needs an infinite family + proof). The genuine finite sub-problem (extend/beat A263996) is real but sits on a wall: the mid-range records were set by exactly the tools we'd bring — simulated annealing / local search on a large distributed compute pool across 1,639 contest entries specifically tuned for \"sophisticated heuristics.\" That is the R(5,5)-witness situation inverted: the record-holder already ran our attack, harder and longer, and the optimal sets are highly structured smooth-number sets ({2^a·3^b·5^c…}) where the search is well-understood and saturated. Worse for a certified-optimality angle: the domain is unbounded integers, so proving a(n) exactly (the DRAT-nonexistence analog) requires first bounding the largest element of an optimal set — a mini-theorem, not clean finite exhaustion/SAT/MIS. Residual movability only on soft, non-contest intermediate terms, at negligible payoff.","fit_score":4,"impact_score":3,"impact_reason":"Sum-product is a marquee problem (huge field significance, $250 prize), but the ONLY finite-computable output — nudging one OEIS A263996 term — is a b-file edit that does not move the asymptotic exponent, is not prize-eligible, and carries little citable weight. High topic fame, near-zero achievable-contribution significance.","verifier":"Trivial exact checker: given A, compute the multiset of pairwise sums and products, dedupe, take union cardinality. O(n^2) integer arithmetic (bignum since products blow up), microseconds even for n in the thousands. Perfect poly-time exact verifier — the strongest part of the fit.","attack":"If attempted (not recommended): seed from published contest optima, run SA / large-neighborhood local search over sets of smooth numbers {2^a 3^b 5^c 7^d}, perturbing by element swap/rescale with the exact O(n^2) objective; and an ILP over a fixed smooth-number support (binary select n of them, minimize |sum-set ∪ product-set| via indicator constraints) for exact small-n confirmation. ~100-core farm, days per n-band. Realistic outcome: reproduce, not beat, the 2016 contest values.","our_edge":"None beyond generic compute. The verifier is trivially cheap (our usual moat), but the record was set by our exact search paradigm at larger scale by a motivated field, and there is no exact-certification hook (unbounded integer domain) the way there was for R(5,5)/kissing on a fixed lattice.","verdict":"PARTIAL — confirms triage. Headline conjecture is asymptotic/analytic = NONE for finite computation (self-declared unresolvable by finite means; current exponent 1.335 vs target 2). A genuine in-class finite sub-problem exists (minimize |A+A ∪ AA| / max(|A+A|,|AA|), extend OEIS A263996) with an ideal cheap exact verifier, but the frontier is a WALL: the record is held by the same SA/local-search tools we'd deploy, run at contest scale (Al Zimmermann 2016), over an unbounded domain with no exact-nonexistence certificate hook. Not worth a farm; any win is a low-impact OEIS nudge, not the $250 result. Do not pursue.","lane":"exact-backtracking","board_class":"NONE","board_class_reason":"no concrete n, authoritative seed witness, or tracked per-n record is identified; the asymptotic headline is not advanced by an unscoped finite score (R2 fails)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/52","links":{"oeis":["A263996"],"arxiv":[]},"frontier":{"summary":"A263996 small-case table (smallest |A+A u AA| over n-sets); contest-set records"}},{"id":64,"title":"Erdős–Gyárfás 2-power cycle conjecture","prize":"$1000 (erdosproblems.com listing; Erdős originally $100 to prove / $50 to disprove)","statement":"Does every finite graph with minimum degree >= 3 contain a cycle of length 2^k for some k >= 2? (Erdős–Gyárfás, 1995.) Falsifiable by a finite min-degree-3 graph avoiding all power-of-two cycle lengths.","finite_object":"A cubic (or min-degree-3) graph on n vertices avoiding every cycle length 4, 8, 16, 32, ... — either as a counterexample (witness) or certified-absent at fixed n (nonexistence).","current_record":"Any counterexample >= 17 vertices; any CUBIC counterexample >= 30 vertices — exhaustive computer search by G. Royle & K. Markström (~2004) via cubic-graph generation (genreg/nauty-style), verifying all cubic graphs on <= 28 vertices (29 impossible by parity). Sharpest near-miss: four 24-vertex cubic graphs with no C4/C8 whose only 2-power cycle is C16. Bound UNMOVED for ~20 years; all 2023–2026 progress is theoretical (P13-free graphs; Liu–Montgomery for large min degree; Carr 2026 'predominantly cubic' structure).","beatable":"WALL","beatable_reason":"The prize-relevant frontier is a wall on both sides. Proving the conjecture needs a new theorem (NONE-like). The disproof/witness search is unbounded (a counterexample has no size ceiling) and near-certainly null: the conjecture is widely believed, Liu–Montgomery proved it for large min degree AND killed the stronger version, and Carr-2026's 'predominantly cubic' structure further constrains any counterexample — a stronger R(5,5)-style no-witness situation. The one MOVABLE piece — extending Markström's exhaustive cubic verification 2–4 vertices (>=30 -> >=32/>=34) — is feasible but uses the record-holder's own tooling, claims no prize, does not resolve the conjecture, and pushes a number the field no longer tracks. Classic 'computational-looking record already at the wall.'","fit_score":5,"impact_score":5,"impact_reason":"Famous named Erdős conjecture and a clean $1000 prize, but a small prize, pure combinatorics with no real-world relevance, and — decisively — the resolvable-by-us portion is a stale, incremental, non-prize bound extension. Resolving the actual conjecture (high impact) is a theorem problem we cannot touch.","verifier":"For a fixed graph, detect existence of a cycle of each length 2^k <= n. In bounded-degree (cubic) graphs, C4/C8 detection is trivial and C16/C32 detection is poly-time (color-coding / bounded DP). Cost is negligible per graph; almost all graphs are eliminated by the cheap C4 test, so amortized cost is tiny. No DRAT needed — nonexistence is an exhaustive-generation certificate (genreg/snarkhunter enumeration is the proof).","attack":"Only the nonexistence sub-problem is finite/bounded. Constrained-generate connected cubic graphs on n=30,32(,34) with the no-C4/no-C8 filter baked into generation (that class is extremely sparse — only 4 graphs at n=24), then test survivors for C16/C32. Extends the certified lower bound from >=30 to >=32/>=34. Rough compute: n=30 ~1–3 core-days, n=32 tens of core-days on a 100-core farm, n>=36 out of reach. The WITNESS (prize) search is unbounded — no n ceiling — and near-certainly null, so it is not a finite computation.","our_edge":"None beyond generic compute. The bound was set by exhaustive cubic-graph generation — exactly our tool — so we would only be re-running the record-holder's method 2–4 vertices further.","verdict":"PARTIAL — Headline is not finitely resolvable by us: proving the conjecture needs a theorem (NONE), and the disproof/witness search is unbounded and near-certainly null (WALL), mirroring the R(5,5) no-witness outcome with even stronger evidence (Liu–Montgomery + 'predominantly cubic'). The one genuine finite sub-problem is extending Markström's exhaustive cubic verification from the 20-year-old >=30-vertex bound to >=32/>=34 via girth-constrained cubic generation plus a cheap C4/C8/C16 verifier — doable on a 100-core farm but incremental, non-prize-claiming, and built on the record-holder's own method. Real-but-minor contribution at best; not a lead.","lane":"exact-backtracking","board_class":"HEAVY","board_class_reason":"movable claim = exhaustive-generation certificate ('no cubic counterexample <= 30 vertices'); a counterexample itself would be witness-cheap but is believed nonexistent (existence bounty, no frontier)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/64","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"no counterexample: verified for all cubic graphs <= 28 vertices (Royle-Markstrom ~2004); extension = exhaustive-generation certificate"}},{"id":67,"title":"Erdős Discrepancy Problem","prize":"$500 — but ALREADY CLAIMED by Tao (problem PROVED). No prize is obtainable by any finite computation.","statement":"For f:ℕ→{−1,+1}, for every C there exist d,m with |∑_{k≤m} f(kd)| > C. Headline PROVED by Tao (2015/16). Finite sub-problem: exact maximal length of a ±1 sequence whose every homogeneous-AP partial sum stays within [−C,C].","finite_object":"A ±1 sequence of length n with discrepancy ≤ C (every sum over {d,2d,…,md} bounded by C in absolute value). Verifier: check all (d,m) with dm≤n — O(n log n) exact, trivial.","current_record":"C=1: 11. C=2 (general): exactly 1160, tight (1161 UNSAT) — Konev & Lisitsa 2014, SAT + ~13GB DRUP cert. C=3 (GENERAL): OPEN — a length-130,000 unrestricted witness published by Konev–Lisitsa has been exactly re-verified by the P42 package; the exact maximum remains unknown. C=3 (multiplicative only): exactly 127,645 (E(3)=127,646), tight — Konev–Lisitsa; this restricted subcase is distinct from the unrestricted frontier. Headline asymptotic result: Tao 2016.","beatable":"MOVABLE","beatable_reason":"Split verdict, R(5,5)-style. (1) WITNESS/lower-bound side is MOVABLE: the longest-known GENERAL discrepancy-3 sequence has sat at ~14,000 since 2014 on Lingeling/Glucose-era solvers. Modern Kissat/CaDiCaL + SLS (YalSAT) + streamliners (the same 'walters' streamliner tricks that took the MULTIPLICATIVE C=3 case to 127k) on a 100-core farm can very plausibly push the general witness to tens of thousands. Beating a concrete number with a trivial exact verifier — clean fit. (2) EXACT-VALUE/tight-bound side is a WALL: proving UNSAT at threshold+1 for general C=3 is the killer. The C=2 UNSAT at 1161 already produced a 'Wikipedia-sized' proof; the general C=3 threshold is far larger (≥14k, likely ≥100k), so its DRAT certificate is almost certainly beyond feasibility — exactly the R(5,5) upper-bound situation. So the tight value is unattainable; only the lower bound moves. And critically: the movable side is LOW-VALUE (see impact).","fit_score":7,"impact_score":3,"impact_reason":"Prize is gone (Tao settled the affirmative). Extending the general discrepancy-3 witness past 14k is a genuine computation but a niche curiosity now that the headline theorem is proven — not a maintained leaderboard, no new asymptotic content, no real-world relevance. The one prestigious deliverable (exact tight value E(3) for general sequences with a DRAT nonexistence cert) is the wall side and is out of reach. Low field significance.","verifier":"Exact and near-free: given a candidate sequence of length n, verify max over all d≤n of running prefix-sums along the d-subsampled sequence stays ≤C. O(n log n) integer arithmetic; DRAT for any UNSAT (nonexistence) step. Canonical verifier-first fit.","attack":"Encode discrepancy-≤3 as CNF: n Boolean vars (signs), cardinality/sequential-counter constraints |partial sum|≤3 over every homogeneous AP {d,2d,…,md}, dm≤n. For the WITNESS: run parallel portfolio (Kissat + CaDiCaL + SLS YalSAT) with streamliners (fix a multiplicative-like or periodic skeleton to seed structure, as Konev–Lisitsa's 'walters(k)' did) to find sat sequences at n = 20k, 40k, 80k. Incremental/assumption-based solving to climb n. Rough compute: witness-finding is embarrassingly parallel; days-to-weeks of a ~100-core farm to push well past 14k — but no tight closure. UNSAT at the true threshold is NOT attempted (infeasible DRAT).","our_edge":"None beyond generic compute + a standard modern SAT/SLS stack. We hold no special lever here; any team with Kissat and a cluster could do it, and the SAT community already has the streamliner know-how (Cornell/Konev). The reason it is un-pushed is disinterest (theorem proved), not difficulty.","verdict":"PARTIAL. Headline is PROVED/asymptotic (NONE by itself), but a genuine in-class finite sub-problem exists: exact maximal length of a general discrepancy-C sequence. C=2 is closed at 1160; C=3 general is OPEN with only a ~14,000 witness lower bound from 2014. Our machinery fits cleanly (trivial O(n log n) verifier, SAT/SLS search, DRAT) and the WITNESS/lower-bound record IS movable with a modern solver farm. BUT: (a) the $500 prize is already claimed and unobtainable; (b) the movable side is low-impact niche work; (c) the high-value side — the exact tight value with a nonexistence certificate — is a WALL (DRAT proof at the true threshold is infeasible, R(5,5)-upper-bound style). Recommendation: NOT worth a campaign. If touched at all, only as a cheap farm shakedown to bump the longest-known general discrepancy-3 sequence, with no expectation of a tight/prize result.","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = +-1 sequence; O(N log N) integer prefix-sum scan; exactly re-verified 130,000-term frontier witness-extendable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/67","campaign_finding":"2026-07-12 correction: the unrestricted Konev-Lisitsa length-130,000 witness was recovered from the authors' archived result artifact and exactly re-verified by the P42 verifier. Earlier ~14,000 frontier and search-cost prose in the originating audit is superseded; new work must start above 130,000.","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"C=3 general witness >= 130,000 (Konev-Lisitsa unrestricted witness, exactly re-verified by P42); C=2 tight at 1160"},"p42_slug":"edp-c3-longest-sequence"},{"id":77,"title":"lim R(k)^{1/k} (diagonal Ramsey growth constant)","prize":"$250","statement":"Determine lim_{k→∞} R(k)^{1/k}, where R(k)=R(k,k) is the diagonal Ramsey number. Currently only known to lie in [√2, <3.8-base]; the limit is not even proven to exist.","finite_object":"A 2-coloring of K_n with no monochromatic K_k (Ramsey graph) certifying a lower bound at a specific order — the finite witness object. But no such object at any single order n moves the asymptotic base, which is the prize.","current_record":"liminf R(k)^{1/k} ≥ √2 ≈ 1.4142 (Erdős 1947 probabilistic lower bound, still the diagonal base); limsup ≤ 3.8^{1+o(1)} base (Gupta-Ndiaye-Norin-Wei, refining Campos-Griffiths-Morris-Sahasrabudhe 2023 \"book algorithm\" — first exponential improvement over Erdős-Szekeres 4^k since 1935). Existence of the limit is OPEN. Feeding small case: R(5,5) ∈ [43 (Exoo 1989), 46 (Angeltveit-McKay 2024)].","beatable":"WALL","beatable_reason":"Headline is a pure asymptotic constant — no finite object determines it; even proving the limit exists needs a new theorem. The only in-class finite sub-problem (exact small R(k,k) / Ramsey-graph constructions) is (a) analytically disconnected from the asymptotic base — pinning R(5,5) or R(6,6) contributes zero digits to the limit, as Erdős himself noted — and (b) itself at the wall: R(5,5) lower bound 43 (Exoo 1989) has stood 37 years, our own R(5,5) witness campaign closed with no witness (42 DRAT-certified asymmetry certs), and the recent upper-bound motion came from LP+casework in the same McKay nauty/SAT ecosystem we'd use. Record set by our own class of tools = classic feasibility-limit signature.","fit_score":3,"impact_score":3,"impact_reason":"The headline is one of the most famous constants in combinatorics (huge field significance), but the prize is only $250 AND is unclaimable by finite computation; the finite contribution we could actually make (an incremental small-Ramsey bound) is low-value and, in the R(5,5) case, already exhausted by us. Real-world relevance nil.","verifier":"O(n^k) / clique-check on a fixed graph: verifying a candidate Ramsey graph of order n has no monochromatic K_k is polynomial in n for fixed k (enumerate k-subsets). Exact and cheap per-order; this is exactly our R(5,5) verifier. But it certifies one small value, not the limit.","attack":"If one wanted the finite slice: SAT/CDCL search for a Ramsey graph beating the current R(k,k) lower bound (e.g. order-43 K5-free symmetric coloring), with DRAT nonexistence certs for the negative direction — precisely our R(5,5) pipeline on a ~100-core farm. But we already ran this: no witness at 43, wall confirmed. Higher k is exponentially further out. Nothing here claims the $250 prize regardless of compute.","our_edge":"Perfect mechanism match to our R(5,5) SAT+DRAT/MIS farm — but no advantage that matters, because the movable-frontier-that-would-count does not exist and the finite sub-frontier is one we already found closed. None beyond generic compute for the prize.","verdict":"PARTIAL-but-WALL. Headline lim R(k)^{1/k} is NONE — a pure asymptotic constant not known to exist, needing a new theorem (√2 ≤ L ≤ 3.8-base). The genuine in-class finite sub-problem (exact small Ramsey numbers / clique-free graph witnesses) maps perfectly to our SAT+DRAT farm but is (1) analytically disconnected from the $250 prize and (2) already at the feasibility wall — our own R(5,5) campaign closed with no witness (42 DRAT asymmetry certs), Exoo's 43 lower bound is 37 years old, and recent motion came from the same McKay tooling we'd use. Do not attack for the prize; the finite slice is a solved/closed wall for us. Skip.","lane":"wall","board_class":"NONE","board_class_reason":"asymptotic limit only; the finite R(5,5) frontier object is carried by #1029 (R4 dedupe)","wall_reason":"asymptotic limit only; the finite R(5,5) frontier object is carried by #1029 (R4 dedupe)","erdos_url":"https://www.erdosproblems.com/77","links":{"oeis":[],"arxiv":[]}},{"id":83,"title":"EKR t=2 intersecting family bound on [4n]","prize":"$500 (already awarded/solved)","statement":"For a family F of 2n-subsets of [4n] with pairwise intersection ≥ 2, |F| ≤ (1/2)(C(4n,2n) − C(2n,n)²). Conjectured by Erdős–Ko–Rado (1961); the t=2 case of the complete intersection theorem.","finite_object":"Per fixed n, the extremal family: all 2n-subsets of [4n] containing ≥ n+1 of the first 2n elements. It is an explicit named construction, NOT the output of any search.","current_record":"PROVED — Ahlswede & Khachatrian, Eur. J. Combin. 18 (1997), resolving Frankl's conjecture; #83 is the t=2/k=2n/m=4n case, RHS = OEIS A071799 (closed form).","beatable":"WALL","beatable_reason":"The headline is a proved theorem with an EXPLICIT extremal construction and a CLOSED-FORM value (A071799). There is no open number to push, no record to beat, and no nonexistence to certify — the maximum and its maximizers are known for all n simultaneously. The imagined \"finite sub-problem\" (small-case exact values / extremal family per n) is already fully determined analytically; a max-independent-set or exhaustion run would only re-derive a theorem for a handful of tiny n before hitting the combinatorial wall, contributing nothing publishable. This is the opposite of the R(5,5) lower-bound witness case: there the extremal object was unknown and searchable; here it is a named formula.","fit_score":2,"impact_score":2,"impact_reason":"Central, celebrated result in extremal set theory (complete intersection theorem), but it is fully solved — no prize is claimable and no frontier remains. A closed sub-problem offers zero marginal contribution.","verifier":"Poly-time count-and-check of a candidate family (verify sizes, pairwise intersections ≥ 2, count). Exact maximality would need max-independent-set on the conflict graph (vertices = 2n-subsets, edges = intersection ≤ 1). Cost blows up: n=3 → 924 vertices, n=4 → 12870 vertices; and the answer is already a proved theorem, so the MIS solve certifies nothing new.","attack":"None worthwhile. The only computational task — MIS/exhaustion on the intersection-conflict graph to reconfirm |F_max| for n=1,2,3 — is a redundant sanity check of a solved theorem, feasible only up to ~n=3–4 and yielding no new value. No SAT/ILP/DRAT deliverable exists because there is no contested boundary.","our_edge":"none beyond generic compute","verdict":"NONE — PROVED theorem (Ahlswede–Khachatrian 1997) with explicit extremal family and closed-form value (A071799). No record to beat, no nonexistence to certify, no movable number. The finite object per n is a named construction, not a search target. Not attackable by our verifier-first farm.","lane":"wall","board_class":"NONE","board_class_reason":"proved theorem (Ahlswede-Khachatrian 1997); a MIS solve certifies nothing new","wall_reason":"proved theorem (Ahlswede-Khachatrian 1997); a MIS solve certifies nothing new","erdos_url":"https://www.erdosproblems.com/83","links":{"oeis":["A071799"],"arxiv":[]}},{"id":86,"title":"C4-free subgraphs of the hypercube (Turán density 1/2?)","prize":"$100 (attaches ONLY to the asymptotic conjecture, which is NONE — unclaimable by finite computation)","statement":"Is every subgraph of Q_n with ≥ (1/2 + o(1))·n·2^{n-1} edges forced to contain a C_4? Equivalently, is the Turán density of C_4 in the hypercube exactly 1/2?","finite_object":"A C_4-free subgraph of Q_n with a record number of edges (or a DRAT certificate that no C_4-free subgraph with k+1 edges exists), pinning ex(Q_n,C_4) = A245762.","current_record":"ex(Q6,C4)=132 proven exact (Harborth–Nienborg, ILP). ex(Q7,C4)≥304 and ex(Q8,C4)≥680 — new LOWER bounds only, set May 2026 by arXiv:2603.29127 via two-phase simulated annealing (~2×10^8 steps). Upper bound / exact value for Q7 and Q8 is OPEN. Asymptotic density pinned in [1/2, 0.6068] (flag-algebra upper bound).","beatable":"UNKNOWN","beatable_reason":"Headline (asymptotic density) is NONE — needs a new theorem, not a witness. The finite sub-problem (exact ex(Q7,C4)) has two frontiers: (1) beating the LB record 304/680 is WALL-ish — it was just set by heavy 2026 simulated annealing, the same tool we'd use, so no edge. (2) The genuinely novel, unattempted target is DRAT-certifying exactness (SAT-UNSAT at 305 edges ⟹ ex(Q7,C4)=304, the first proven term past n=6). Feasibility is UNKNOWN: the paper's own ILP failed to prove the Q7 upper bound, and |Aut(Q7)|=645,120 symmetry helps but may not tame it on a ~100-core farm; Q8 almost certainly a wall.","fit_score":5,"impact_score":3,"impact_reason":"Prize is unreachable (asymptotic). A DRAT-proven ex(Q7,C4) would be a real but modest extremal-combinatorics / OEIS-A245762 extension — the first exact value past n=6 — with no real-world relevance and limited field weight.","verifier":"C_4-freeness check: enumerate the fixed 4-cycles of Q_n (672 in Q7, 1792 in Q8) and verify none is fully contained; O(#cycles) per candidate, microseconds. Trivially exact and cheap.","attack":"SAT feasibility/optimality on Q7: 448 boolean edge vars, totalizer cardinality \"sum ≥ 305\", 672 length-4 clauses (one per 4-cycle: at least one edge absent), plus lex-leader symmetry breaking over Aut(Q7)=Z2^7⋊S7 (645,120). SAT at 305 ⟹ new record; UNSAT+DRAT ⟹ ex(Q7,C4)=304 proven, extending A245762. In parallel run ILP (maximize Σx_e s.t. per-cycle ≤3) and portfolio local-search/SA to probe 305–310. Rough compute: Q7 instance is small (sub-R(5,5)); a few core-days to weeks of CDCL if it closes, but could be an unbounded wall if the UNSAT is hard. Q8 out of reach.","our_edge":"The exactness/DRAT certificate on Q7 is the R(5,5)-style find-or-certify move that nobody in the literature has attempted (all prior work reports only SA/ILP lower bounds); our verifier-first SAT+DRAT + symmetry-breaking + CPU-farm pipeline is exactly built for it. No edge on the SA lower-bound record itself (already saturated by fresh 2026 SA).","verdict":"PARTIAL. Headline asymptotic conjecture is NONE (needs a new theorem; density stuck in [1/2, 0.6068]). But a real in-class finite sub-problem exists: the exact extremal numbers ex(Q_n,C4)=A245762, proven only through n=6 (=132); Q7≥304 and Q8≥680 are 2026 lower bounds with OPEN optimality. Our best shot is the unattempted R(5,5)-style play — SAT+DRAT to certify ex(Q7,C4)=304 exactly (or find 305), extending the OEIS table. Movability is UNKNOWN (prior ILP failed on the Q7 upper bound), and it claims no prize. Worth a scoped Q7 SAT probe, not a flagship campaign.","lane":"SAT+DRAT-nonexistence","board_class":"READY","board_class_reason":"witness = edge subset of Q7; scan the 672 4-cycles, exact, microseconds; ex(Q7,C4) >= 305 witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/86","links":{"oeis":["A245762"],"arxiv":["2603.29127"]},"frontier":{"summary":"ex(Q7,C4) >= 304, ex(Q8,C4) >= 680 (SA, arXiv:2603.29127, May 2026); exact values open past ex(Q6,C4)=132"},"p42_slug":"hypercube-q7-c4-free"},{"id":90,"title":"Unit distance problem — max unit pairs (A186705)","prize":"$500 (headline) — already effectively resolved: DISPROVED in the negative, Lean-verified, by an OpenAI internal model. Earlier $300/$250 sub-offers were for the asymptotic n^{1+o(1)} bound, now settled. No prize attaches to the finite OEIS-extension sub-problem.","statement":"Erdős #90: does every set of n distinct points in R^2 have at most n^{1+O(1/loglog n)} pairs at distance exactly 1? Headline asymptotic conjecture (best classical upper bound O(n^{4/3}), Spencer–Szemerédi–Trotter 1984). Finite sub-problem: OEIS A186705, u(n) = max number of unit-distance pairs among n points.","finite_object":"A 22-point configuration in R^2 maximizing the number of unit-distance pairs (equivalently, the abstract maximally-dense unit-distance graph on 22 vertices), which would establish u(22) and extend the OEIS A186705 exact table by one term.","current_record":"Exact values u(n) proven for n ≤ 21 (u(21)=57); Alexeev, Mixon, Parshall, arXiv:2412.11914 (2024/2025), via forbidden-subgraph filtering + totally-unfaithful-graph elimination + custom embeddability solver, ~6,100 CPU-hours. Prior: n≤14 (Schade 1993), n=15 (Ágoston–Pálvölgyi), n=16–17 extended. Next open value: u(22). Densest-known lower bounds up to n=100 via diverse beam search, arXiv:2406.15317 (2024), 60M-UDG database. Headline asymptotic conjecture DISPROVED (negative, Lean-verified) by an OpenAI internal model.","beatable":"WALL","beatable_reason":"Two decisive blocks against our verifier-first farm. (1) The exact-value verifier is not poly-time: deciding graph→unit-distance-realizability in R^2 is ∃R-hard, violating our core precondition; the specialists' own solver is incomplete (aborts on girth-≥5 non-UDGs, which dominate at n≥22). (2) Already at feasibility limit set by the same class of tools: n=22 needs ~15,000 CPU-hours (compute within reach) but the binding constraint is the realizability oracle, not cycles — our generic SAT/ILP/MIS/SA toolbox supplies no such certificate, and we'd have to out-engineer the specialists' incomplete geometric solver. The one movable sub-lane (beat a densest-known lower bound for some 30= k of the others equidistant from it (i.e. lying on a common circle centered at that point). Minimal such n is g(k); the extremal table {g(3)=6, g(4)=8, g(5)=?} is the finite record.","current_record":"Headline f(n)<=n^{o(1)}: DISPROVED (negative) — it strengthens the unit-distance conjecture #90, so the 2026 disproof of #90 disproves it; $500 was for the now-false upper bound, $50/$100 for the counterexample. Finite table: g(3)=6, g(4)=8, computed by Fishburn (Erdős–Fishburn, \"Minimum planar sets with maximum equidistance counts,\" Comput. Geom. 7(4):207–218, 1997). No extension to g(5) in ~30 years; no 2023–2026 movement found.","beatable":"WALL","beatable_reason":"The exact-value record g(5) is a real-algebraic wall, not a discrete search. The construction (upper-bound) side has a trivial verifier but lives in continuous R^{2n}, so our SAT/ILP/MIS/exhaustion farm only applies under a lattice/algebraic restriction, yielding non-rigorous best-known constructions — not the exact value, and not a prize (prize is dead). The nonexistence direction (\"no (n-1)-point set achieves f=k\") is a statement over the reals: realizability of a point/circle incidence structure in the plane is ER-complete, with no finite discrete/DRAT encoding; CAD is the honest tool and does not scale. This is the R(5,5)-upper-bound-style machinery mismatch, and Fishburn's own computation stalled at k=4 for 30 years.","fit_score":2,"impact_score":3,"impact_reason":"Prize is effectively dead: the $500 upper-bound conjecture is disproved and the counterexample bounty ($50) is subsumed by the #90 disproof. Distance problems are central Erdős combinatorics, but this specific max-equidistance-count table is a footnote that has drawn no activity in 30 years. No real-world relevance.","verifier":"Construction side only: given exact rational/algebraic coordinates of n points, compute all C(n,2) squared distances and check each point has some distance value with multiplicity >= k. O(n^2) exact, trivial. This verifies UPPER bounds on g(k) (constructions). NO finite verifier exists for the nonexistence/exact-value direction.","attack":"Only the soft side is tractable: ILP/local-search/SA to select subsets of triangular or square lattices (or small algebraic point families) maximizing the min equidistance count, with exact rational snapping, giving improved CONSTRUCTIONS = upper bounds on g(k) / lower bounds on f(n) for specific n. Modest CPU (single node, hours). But this cannot certify an exact g(5): ruling out all real (n-1)-point configs needs cylindrical algebraic decomposition over R, which is infeasible past tiny n. No DRAT-style nonexistence certificate is possible.","our_edge":"None beyond generic compute. Our verifier-first discrete farm has no purchase on the ER-complete nonexistence direction, and the construction side is a continuous-geometry optimization where our tooling offers no specific advantage over standard geometric search.","verdict":"PARTIAL classification, but WALL for the movable record. Headline is a disproved asymptotic conjecture (NONE for finite computation). The genuine finite sub-problem is the Erdős–Fishburn table g(k) = smallest n with f(n)=k (g(3)=6, g(4)=8). Its exact-value frontier (g(5)) is unreachable by our SAT/ILP/MIS/DRAT farm: the nonexistence direction is ER-complete over the reals with no discrete certificate, and the only tool-tractable side (lattice construction search) yields non-rigorous, non-prize best-known bounds. Poor fit; do not pursue.","lane":"wall","board_class":"NONE","board_class_reason":"headline disproved with #90; exact-value direction has no finite verifier; equality-constrained coordinates (R3 fails)","wall_reason":"headline disproved with #90; exact-value direction has no finite verifier; equality-constrained coordinates (R3 fails)","erdos_url":"https://www.erdosproblems.com/92","links":{"oeis":[],"arxiv":[]}},{"id":97,"title":"Convex polygon: every vertex has 4 equidistant others","prize":"$100","statement":"Does every convex polygon have a vertex with no other 4 vertices equidistant from it? A counterexample is a convex polygon where every vertex has >=4 other vertices at a common distance (>=4 others on a circle centered at it). Erdos conjectured yes (some vertex always fails); disproof = one finite counterexample.","finite_object":"A convex polygon (point set in the plane, ideally exact rational/algebraic coords) in which every vertex has at least 4 other vertices at a single common distance.","current_record":"k=3 fully achieved (conjecture false at 3): Danzer convex 9-gon (per-vertex varying distance) and Fishburn-Reeds 1992 convex 20-gon (uniform unit distance). k=4: NO example of any size known; open. AlphaEvolve (Google DeepMind, arXiv:2511.02864, Nov 2025) attacked this exact problem with an exact/interval-arithmetic verifier and reproduced only k=3, failing to reach k=4.","beatable":"WALL","beatable_reason":"The applicable search is continuous/algebraic local search over real vertex coordinates, and the SOTA instance of exactly that tool class (AlphaEvolve: exact verifier + Google-scale evolutionary/LLM search) targeted this precise problem in 2025 and failed to exceed the already-known k=3. It does NOT reduce to our discrete farm: no size parameter to exhaust, so SAT/ILP/max-independent-set/exhaustion don't apply; a k=4 witness lies on a measure-zero algebraic variety (many exact concyclic-equal-distance quadruples + convexity at once) reachable only by a symbolic/structural construction (a new idea, not more CPU). And Erdos's conjecture direction (nonexistence) is uncertifiable by finite computation -- unbounded vertex count, continuous coords, no DRAT/finite-size analogue -- so the R(5,5)-style certify-no-witness route is structurally unavailable.","fit_score":3,"impact_score":3,"impact_reason":"Only a $100 prize; a genuine but minor Erdos problem in combinatorial geometry. Mild field interest, some visibility from the AlphaEvolve case study, but no real-world relevance and low mathematical stakes relative to Ramsey/kissing records.","verifier":"Given rational vertex coords: (a) verify convex position (O(n) orientation checks) and (b) for each vertex, group squared distances to others and check some value has multiplicity >=4 (sort per vertex). Exact, O(n^2 log n). Cheap and clean.","attack":"If it were movable: simulated annealing / evolutionary search over vertex coords minimizing per-vertex max equidistance error, normalized by min side length, then snap to exact algebraic coordinates. This IS AlphaEvolve's approach and it failed at scale. The only credible route is symbolic -- pick a combinatorial concyclic-quadruple incidence pattern, set up the polynomial system for exact equal distances + convexity, and solve via Groebner bases / CAD -- which is symbolic algebra, not our SAT/MIS/SA farm, with severe combinatorial blowup and no success guarantee. Rough compute: unbounded / no bounded search space.","our_edge":"None beyond generic compute -- and generic verifier-first evolutionary compute (AlphaEvolve-class) already ran this exact problem at Google scale in 2025 and could not exceed k=3. Our SAT/ILP/MIS/DRAT edge (discrete, bounded-size, certifiable-nonexistence) does not transfer to this continuous/algebraic unbounded-size existence question.","verdict":"PARTIAL-leaning-NONE, WALL. Falsifiable finite existence with a clean exact poly-time verifier (the only axis that fits), but NOT our machinery: the witness hunt is continuous/algebraic with no numeric frontier to push and no bounded size, SAT/ILP/MIS/exhaustion don't apply, the nonexistence direction is uncertifiable by finite computation, and the SOTA verifier-first search (AlphaEvolve, 2025) already tried this exact problem and failed to beat the known k=3. Not STRONG. Recommend DROP -- no compute lever we own moves this frontier.","lane":"wall","board_class":"NONE","board_class_reason":"existence bounty with no numeric frontier, and the defining condition (4 equidistant points) is equality-constrained real geometry (R3 fails: rational restriction lossy)","wall_reason":"existence bounty with no numeric frontier, and the defining condition (4 equidistant points) is equality-constrained real geometry (R3 fails: rational restriction lossy)","erdos_url":"https://www.erdosproblems.com/97","links":{"oeis":[],"arxiv":["2511.02864"]}},{"id":101,"title":"Erdős #101 — 4-point lines o(n²), orchard A006065","prize":"$100","statement":"Given n points in R² with no five collinear, the number of lines containing exactly four points is o(n²). Asymptotic upper-bound conjecture; $100. Associated finite object: OEIS A006065, max number of 4-point lines (orchard, 4-per-row).","finite_object":"A point configuration of n points in R² (rational coords), no 5 collinear, maximizing the count of lines with exactly 4 points — OEIS A006065 (real) / A172992 (integer-coordinate variant).","current_record":"Exact optimal proven to n=20, a(20)=23 (Zhao Hui Du, exhaustive search, 2008–2010). Lower bounds beyond: a(22)≥28 (Du); Ed Pegg n=22 construction (2020); integer variant A172992 to n=27 (Kamenetsky, 2012). No exact-value progress since 2010.","beatable":"WALL","beatable_reason":"The prize headline is asymptotic (o(n²), no 5 collinear) — not finite-resolvable; a single config can't disprove it. The genuine finite frontier — extending exact/optimal A006065 values past n=20 — requires enumerating super-exponentially many order types AND certifying geometric realizability in R² (pseudoline stretchability, ∃R-complete/NP-hard). That is a real-algebraic realizability oracle, NOT a Boolean UNSAT/DRAT problem, so it does not map to our SAT/ILP/DRAT farm; it is the R(5,5)-upper-bound analog and has been stuck 16 years. The only tool-fittable crack — bumping a b-file lower bound (integer-grid SA/ILP, e.g. A172992) — is low-significance (integer optima are provably below the real optimum for several n), already worked by hobbyist grid search with no headroom, and the strong large-n records are algebraic (cubic-curve group law) that generic continuous/discrete search cannot out-produce.","fit_score":3,"impact_score":3,"impact_reason":"$100 prize attaches only to the asymptotic headline we cannot touch. The orchard 4-point variant is a classic, cute, OEIS-\"nice\" combinatorial-geometry object but niche; no real-world relevance. A b-file/OEIS lower-bound extension would be a minor, non-prize contribution.","verifier":"Exact rational determinant collinearity checker: for a candidate n-point set compute all C(n,4) collinearity tests (or sort slopes per point), count size-exactly-4 lines, assert no line has ≥5. O(n^4), microseconds for n≤30. Poly-time, cheap.","attack":"Exact-value direction (the real record): infeasible — order-type enumeration ~10^13+ by n≈13 and worse at n=21, plus a real-realizability oracle per type; DRAT-style nonexistence certs do not apply to ∃R. Lower-bound direction: SA/ILP over an integer grid maximizing exactly-4-point lines subject to no-5-collinear, seeded from cubic-curve constructions; cheap (~100 core-hours) but very unlikely to beat algebraic records and would only nudge a b-file, not claim the prize or an optimality record.","our_edge":"None beyond generic compute. Our cheap exact verifier is well-suited, but the movable side needs a realizability certificate (∃R), which our SAT+DRAT / MIS machinery does not produce; the record constructions are algebraic-geometry-driven.","verdict":"PARTIAL — genuine finite in-class sub-problem (A006065 exact values, an OEIS hard/nice/more extension target) exists, but for OUR toolkit it is a WALL: exact-value extension is a geometric-realizability (∃R) problem unsuited to SAT/ILP/DRAT and stuck since 2010; the only tool-fittable direction (lower-bound b-file bumps) is low-value and already at its feasibility limit against algebraic records. Recommend PASS.","lane":"wall","board_class":"NONE","board_class_reason":"collinearity is an equality constraint: optimal configurations may require irrational coordinates (ExistsR realizability gap), so a rational-witness board is not lossless (R3 fails); nonexistence side has no DRAT route","wall_reason":"collinearity is an equality constraint: optimal configurations may require irrational coordinates (ExistsR realizability gap), so a rational-witness board is not lossless (R3 fails); nonexistence side has no DRAT route","erdos_url":"https://www.erdosproblems.com/101","links":{"oeis":["A006065","A172992"],"arxiv":[]}},{"id":107,"title":"Erdős–Szekeres \"Happy Ending\" convex-polygon conjecture","prize":"$500","statement":"f(n) = least N such that any N points in R^2 in general position (no 3 collinear) contain n points in convex position. Conjecture: f(n)=2^{n-2}+1. Verified n≤6 (f=3,5,9,17); f(7) open, conjectured 33.","finite_object":"A point configuration: either (falsify) 33 points in general position with NO convex 7-gon, or (prove f(7)=33) a DRAT-certified UNSAT showing no such 33-point order type exists.","current_record":"f(7) unknown, bracketed 33 ≤ f(7) ≤ 127. Lower 33 = Erdős–Szekeres 1935 construction (2^{n-2}, tight for n≤6). Upper 127 = Tóth–Valtr 2005 (C(2n-5,n-2)+1). Last exact value f(6)=17: Szekeres–Peters 2006 (~1500–3000 GHz-hrs), SAT-reproven to 8.53 CPU-sec by Heule–Scheucher 2024. Latest n=7 SAT attack: Bogdan, arXiv 2512.24061, Dec 2025 (partial, blocked). Asymptotic upper bound improved at SoCG 2025 (Baek–Balko).","beatable":"WALL","beatable_reason":"The one live finite frontier is f(7)=33. It is being attacked RIGHT NOW with our exact machinery — order-type/triple-orientation SAT + DRAT certificates — by the world leaders (Heule, Scheucher, Marić, Bogdan). Bogdan's Dec-2025 SAT attack (16.67M clauses) only closes anchored subfamilies and hits heavy-tailed runtimes up to 2.28M CPU-sec per config, explicitly \"far from resolving ES(7)=33.\" We have no edge over the people who own the ES-SAT encoding, and a 100-core farm cannot dent heavy-tailed n=7 instances. Falsification search (33 points, no convex heptagon) targets an object almost certainly nonexistent (construction tight for all n≤6) → expected null with no certificate. The $500 headline (all n) is a pure theorem.","fit_score":6,"impact_score":6,"impact_reason":"Prize weight low ($500), but field significance very high: a 90-year-old Erdős flagship (\"Happy Ending problem\"), and settling f(7) would be the first new exact value in 20 years — a headline result in discrete geometry. Real-world relevance negligible.","verifier":"Convexity check: a k-set is in convex position iff all its 4-subsets are convex (poly-time). Checking a 33-set contains a convex 7-gon = scan C(33,7)≈4.3M 7-subsets (or via the order type) — cheap, exact, DRAT-certifiable in the SAT encoding. Excellent verifier; this is exactly why the form maps to our farm.","attack":"SAT-encode 33 points via signotope/triple-orientation axioms + clauses forbidding any convex 7-subset; solve for UNSAT (⇒ f(7)≤33, DRAT cert) or a realizable model (⇒ counterexample). Falsification alternative: simulated-annealing/local-search over 33-point real coordinates minimizing #convex-heptagons, seeking 0. Compute: intractable at n=7 — Bogdan's anchored subfamilies alone reach 2.28M CPU-sec with heavy tails; the full instance is far beyond a 100-core farm, and SA search is expected to return null (construction tight).","our_edge":"None beyond generic compute. Heule/Scheucher/Marić/Bogdan already run the identical SAT+DRAT order-type pipeline we would use and have optimized it for years (f(6) 3000 GHz-hrs → 8.53 s). No proprietary encoding, no verifier moat, no compute advantage at the heavy-tailed n=7 scale.","verdict":"PARTIAL — the $500 headline (prove f(n)=2^{n-2}+1 for all n) is a pure theorem (NONE), but there is a genuine finite in-class sub-problem: the exact value f(7) (conjectured 33) and the falsification search for a 33-point set with no convex 7-gon, both with a cheap exact verifier that maps perfectly onto our SAT+DRAT template. However, that sub-problem is a WALL for us: the world's best are already running our exact tools on it and are blocked by heavy-tailed runtime, and we hold no edge. Form fits beautifully; frontier is walled. Do not farm.","lane":"SAT+DRAT-nonexistence","board_class":"HEAVY","board_class_reason":"orientation checking is exact, but no a-priori coordinate bit bound is established; a byte-capped integer format could exclude valid order types (R1/R3 admission proof pending)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/107","links":{"oeis":[],"arxiv":["2512.24061"]},"frontier":{"summary":"33 <= f(7) <= 127 (a 33-point general-position set with no convex 7-gon would refute Erdos-Szekeres at n=7)"}},{"id":114,"title":"Erdős–Herzog–Piranian lemniscate max length","prize":"$250 (Erdős); falsification-only prize (disprove by counterexample).","statement":"Is the arc length of the lemniscate {z∈ℂ:|p(z)|=1} over monic degree-n polynomials maximized by p(z)=z^n−1?","finite_object":"None in our sense. The \"object\" is a monic polynomial in continuous coefficient space ℂ^{n−1}; the frontier is a transcendental arc-length functional, not a discrete combinatorial witness.","current_record":"Conjecture PROVEN for all sufficiently large n (Terence Tao, arXiv 2512.12455, Dec 2025), via Stokes/area-integral analysis; asymptotic bound ℓ ≤ 2n+4log2+o(1). z^n−1 proven a LOCAL maximum (Fryntov–Nazarov). n=2 (Bernoulli lemniscate) proven maximal (Eremenko–Hayman). n=3, n=4 remain OPEN. Conjectured max = B(1/2,n/2)=2n+4log2+O(1/n).","beatable":"WALL","beatable_reason":"Three independent walls. (1) No exact verifier: arc length of {|p|=1} is transcendental (elliptic integrals / beta function); a candidate can only be scored by high-precision numerical curve-tracing, never exactly or DRAT-certified — the core requirement of our farm is absent. (2) The counterexample is essentially ruled out: z^n−1 is a proven local maximum (kills local-search/annealing near the optimum) AND Tao proved the full conjecture for all large n (confines any counterexample to small n where the theory is strongest); AlphaEvolve's own search rapidly reconverged to z^n−1 with no counterexample. (3) The genuine open sub-problems (rigorous n=3, n=4; explicit large-n threshold) are continuous global-optimization/validated-numerics or hard-analysis problems — interval-arithmetic branch-and-bound over coefficient space or new theorems — not SAT/ILP/MIS/exhaustion. Every record here was set by hard analysis, not by tools we would use.","fit_score":2,"impact_score":4,"impact_reason":"Classical, famous Erdős problem freshly in the spotlight (Tao Dec 2025), so field significance is real, but prize is small ($250) and payable only by a counterexample that the strongest available theory says almost certainly does not exist; zero real-world relevance.","verifier":"Compute arc length of the level set {|p(z)|=1}: trace the (possibly multi-oval) curve and integrate ds numerically. NOT poly-time-exact — it is transcendental; rigorous certification would need interval-arithmetic enclosures of an elliptic-type integral over the whole level set. Cost: cheap for a heuristic float estimate, expensive and non-standard for a certified enclosure; no DRAT analog exists.","attack":"The only in-family attack is SA/local-search over monic coefficients maximizing numerically-estimated lemniscate length per small degree (n=3,4,5), hoping to beat ℓ(z^n−1). Rough compute: trivial (hours on a few cores) to run, but near-zero expected value — the objective's global max is a proven local max and the conjecture is proven for large n, so a \"hit\" is a lottery; and any numerical hit could not be exactly certified, only motivate a separate validated-numerics proof (branch-and-bound with rigorous integral enclosures over ℂ^{n−1}), which is a different discipline from our exact-witness farm.","our_edge":"None beyond generic compute. Our exact-verifier-first, discrete-witness/DRAT machinery does not apply to a transcendental continuous-optimization objective; the relevant tooling (validated numerics, complex analysis) is outside our precedent set (kissing/Ramsey/LP-SDP certificates).","verdict":"WALL / poor-fit. Genuinely open at small n (n=3,4) and at the explicit large-n threshold, so not pure-NONE in the abstract, but there is NO exact poly-time verifier (arc length is transcendental), the counterexample the $250 rewards is essentially precluded (z^n−1 is a proven local maximum and Tao proved the conjecture for all large n), and the real open sub-problems require validated numerics or new hard analysis — outside our SAT/ILP/MIS/exhaustion/DRAT precedent. Do not attack: near-zero EV, no certifiable witness path, no edge beyond generic compute.","lane":"wall","board_class":"NONE","board_class_reason":"transcendental objective (elliptic-integral arc length); no exact/DRAT verifier exists; Tao proved the conjecture for large n (Dec 2025)","wall_reason":"transcendental objective (elliptic-integral arc length); no exact/DRAT verifier exists; Tao proved the conjecture for large n (Dec 2025)","erdos_url":"https://www.erdosproblems.com/114","links":{"oeis":[],"arxiv":["2512.12455"]}},{"id":128,"title":"Erdős Sparse Half Conjecture (triangle-free)","prize":"$250","statement":"Must a graph G on n vertices in which every induced subgraph on ≥⌊n/2⌋ vertices has > n²/50 edges contain a triangle? Contrapositive of the Sparse Half Conjecture: every triangle-free graph on n vertices has a ⌊n/2⌋-vertex induced subgraph (\"half\") with ≤ n²/50 edges.","finite_object":"A putative counterexample: a single finite triangle-free graph on n vertices whose SPARSEST ⌊n/2⌋-vertex induced subgraph has strictly > n²/50 edges.","current_record":"Theorem side: Razborov (2021, \"More about sparse halves in triangle-free graphs\", flag-algebra/SDP) proved every triangle-free graph has a half with ≤ (27/1024)n² ≈ 0.02637 n² edges. Erdős-conjectured constant 1/50 = 0.02 n². Proven exactly in special cases: girth ≥5, independence number ≥2n/5 (Krivelevich), strongly regular graphs, and min-degree > 10n/29 / ≥5n/14 (Keevash–Sudakov and successors). Extremal tight examples: uniform blowup of C₅ and blowup of the Petersen graph, each hitting exactly n²/50 in the sparsest half. $250 prize.","beatable":"WALL","beatable_reason":"Three independent walls. (1) NON-POLY VERIFIER: certifying a candidate is a counterexample requires proving its sparsest ⌊n/2⌋-vertex induced subgraph has > n²/50 edges — that is exact Sparsest-k-Subgraph, NP-hard, a min over C(n,n/2) subsets. This directly fails the STRONG poly-time-verifier criterion; SAT/ILP can do it only for tiny n, which is exactly the uninteresting regime. (2) ZERO SLACK + object almost certainly nonexistent: the conjecture is widely believed TRUE, and the known tight constructions (C₅/Petersen blowups) hit the boundary EXACTLY (=n²/50), so a counterexample must strictly beat the best tight construction with no margin — decades of effort plus Razborov's flag-algebra evidence say none exists. (3) WRONG FRONTIER FOR A FARM: the live frontier is closing the analytic gap 27/1024 → 1/50, an SDP/flag-algebra THEOREM problem Razborov already pushed with heavy machinery; the extremal templates are known and a naive CPU-farm search over small weighted triangle-free templates just reproduces the flag-algebra LP, which certifies 27/1024, not 1/50. Small-n exhaustive/SAT search would only re-confirm no counterexample where results already guarantee it.","fit_score":2,"impact_score":5,"impact_reason":"Famous named Erdős conjecture (Sparse Half), a driver of the flag-algebra program and studied for ~50 years, so field significance is real. But the prize is tiny ($250), there is no real-world relevance, and a finite counterexample (which almost certainly does not exist) would upend belief rather than incrementally advance — the credible progress is theorem-side, not witness-side.","verifier":"Exact sparsest-half check = minimize induced edge count over all ⌊n/2⌋-subsets (NP-hard Sparsest-k-Subgraph). Cost: ILP/SAT feasible only for n≲40–60 per candidate; the meaningful regime is asymptotic blowups where the \"verifier\" becomes a fractional quadratic minimization over vertex weights (flag-algebra SDP).","attack":"If one insisted: enumerate triangle-free graphs / weighted triangle-free templates on ≤15–20 vertices, and for each optimize vertex blow-up weights to maximize the min fractional half density, seeking any template with min-half > 1/50 (LP/SDP inner problem per template, ILP for the discrete min-half). Rough compute: hours-to-days on ~100 cores. Expected outcome: NEGATIVE — reproduces C₅/Petersen at exactly 1/50 and Razborov's 27/1024 certified ceiling; no counterexample. Not worth farm time.","our_edge":"None beyond generic compute. Our exact-verifier + CPU-farm edge is neutralized because the verifier is itself NP-hard and the real object is a fractional/analytic extremum where flag-algebra SDP (Razborov) already dominates brute force.","verdict":"WALL / PARTIAL-at-best. The \"FALSIFIABLE by finite counterexample\" tag is technically true but misleading for our farm: (a) the verifier is NP-hard (sparsest-half), violating the poly-time-verifier requirement of STRONG; (b) the target object almost certainly does not exist — the conjecture is believed true, proven in many cases, and the tight examples hit the boundary EXACTLY with zero slack; (c) the genuine live frontier is the analytic gap 27/1024 → 1/50, an SDP/flag-algebra theorem Razborov already advanced, which brute force cannot match. A small-n SAT/ILP counterexample sweep would only re-confirm non-existence. Not a fit for verifier-first witness farming. Recommend PASS.","lane":"wall","board_class":"NONE","board_class_reason":"verifier is NP-hard (Sparsest-k-Subgraph); meaningful regime is flag-algebra SDP blowups, not finite witnesses","wall_reason":"verifier is NP-hard (Sparsest-k-Subgraph); meaningful regime is flag-algebra SDP blowups, not finite witnesses","erdos_url":"https://www.erdosproblems.com/128","links":{"oeis":[],"arxiv":[]}},{"id":138,"title":"van der Waerden numbers W(k)","prize":"$500 (for improving bounds / proving W(k)^{1/k} -> infinity)","statement":"Improve the bounds for the 2-color van der Waerden number W(k) (least N such that every 2-coloring of {1..N} has a monochromatic k-term AP); headline: prove W(k)^{1/k} -> infinity.","finite_object":"A 2-coloring of {1,...,N} with no monochromatic k-term AP (lower-bound witness), or a DRAT UNSAT certificate that none exists at a given N (exact value).","current_record":"Exact: W(2,3)=9, W(2,4)=35 (Chvatal 1970), W(2,5)=178 (Stevens-Shantaram 1978), W(2,6)=1132 (Kouril & Paul 2008, SAT+FPGA). W(2,7) unknown. Best lower bounds: W(2,7)>=3703, W(2,8)>=11495, W(2,9)>=41265, W(2,10)>=103474 (Ahmed et al. 2014 / Kouril / Heule SAT colorings; refined by D. Monroe 2023 SLS). Asymptotic frontier: Green 2021, w(3,k) >= k^{b(k)}, b(k)=c((log k)/log log k)^{1/3}, later sharpened - algebraic, not computational.","beatable":"WALL","beatable_reason":"Two finite sub-problems, both walled. (1) Exact-value extension: only W(2,3..6) known (W(2,6)=1132 needed clusters+FPGAs); W(2,7) ~ 10^4-10^5, so an UNSAT+DRAT cert at that N is astronomically infeasible - identical to the R(5,5) upper-bound wall. (2) Lower-bound record colorings ARE a live SAT/SLS problem with a trivial verifier, but the records are held by exactly the tools we'd use (tuned stochastic local search: Ahmed 2014, Heule, Monroe's 2023 DDFW linear-weight-transfer solver built specifically to crack these instances). No verifier moat (checker is commodity), DRAT-nonexistence infeasible, and any gain is a marginal, non-prize nudge won only by out-tuning SLS specialists. No edge beyond generic compute.","fit_score":6,"impact_score":5,"impact_reason":"Famous, central Erdős/Ramsey-theory quantity with high field significance, but the reachable piece is only a marginal lower-bound record (non-prize), the $500 prize itself requires an unreachable asymptotic theorem, and real-world relevance is negligible.","verifier":"Poly-time exact: scan a candidate coloring for a monochromatic k-AP in O(N^2/k). Trivial and cheap - but it is a commodity every prior record-holder already runs, so no moat.","attack":"Only movable target is a lower-bound record: run farm-scale stochastic local search (DDFW/yalsat with linear weight transfer, or CDCL-with-restarts) on the CNF \"2-color {1..N}, no mono k-AP\" to find a valid coloring at N above the current record for some k; verify by AP scan. Rough compute: farm-weeks of restarts per candidate k, competing head-to-head against purpose-tuned specialist solvers. Exact-value W(2,7) (UNSAT+DRAT) is out of reach at N~10^4-10^5.","our_edge":"None beyond generic compute. The record is held by exactly our toolset (tuned SLS SAT solvers); no unique verifier and no feasible DRAT-nonexistence angle.","verdict":"PARTIAL. Prize/headline (W(k)^{1/k}->infinity) is asymptotic = NONE; the genuine finite sub-problem (record colorings / exact-value extension) exists and maps cleanly onto a SAT-witness farm with a cheap exact verifier, but is practically at the WALL: exact extension infeasible, and lower-bound records are a saturated SLS arms race held by our own tools with no moat and only non-prize gains. Do not pursue.","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = 2-coloring of [N]; O(N^2/k) mono-AP scan, exact; W(2,7) > 3703 witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/138","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"W(2,7) >= 3703 (Ahmed et al. 2014); W(2,6)=1132 exact (Kouril-Paul 2008)"}},{"id":139,"title":"Exact r_k(N): largest AP-free subset table (Erdős–Turán / Szemerédi)","prize":null,"statement":"Let r_k(N) be the size of the largest subset of {1,...,N} with no non-trivial k-term arithmetic progression. Conjecture (Erdős–Turán): r_k(N)=o(N). This headline is PROVED (Szemerédi 1975); the $1000 prize is already claimed. The finite sub-problem in-class is computing EXACT values of r_k(N) — the OEIS tables A003002 (k=3), A003003/4/5 — which is a max-independent-set / SAT extremal search with a trivial exact verifier.","finite_object":"A 3-AP-free (or k-AP-free) subset of {1,...,N} of size K, OR a DRAT nonexistence certificate that no such K-set exists (fixing r_k(N)=K-1).","current_record":"r_3(211)=43, last confirmed OEIS A003002 term (Fausto A. C. Cariboni b-file, n=0..211, exact MIS/backtracking, ~2019–2024). r_3(212) OPEN — conjectured 43, unproven; Ergezer 2026 (arXiv:2606.04016, UMass Unity cluster, ~7,850 worker-hrs) found zero feasible 44-sets but produced no infeasibility certificate. Asymptotic prize itself: claimed by Szemerédi [Sz75]; best bounds Kelley–Meka 2023 (k=3), Green–Tao 2017 (k=4), Leng–Sah–Sawhney 2024 (k≥5) — all asymptotic, not finite.","beatable":"WALL","beatable_reason":"The only meaningful frontier is r_3(212) (r_3(211)=43 is the last confirmed OEIS A003002 term; 212 is either 43 or 44). Ergezer 2026 (arXiv 2606.04016) threw ~7,850 worker-hours on a SLURM cluster at exactly this value using exactly our arsenal — OR-Tools CP-SAT + HiGHS MIP + CaDiCaL CDCL/SAT with depth-d witness splitting — and it RESISTED. The LP relaxation is flat (dual bounds pinned at 0.0 after 8h), and two chunks (T1c) are paradigm-invariant: CP-SAT, MIP, 12h pure CDCL, and windowed CDCL all fail. The author states the certificate is 'too combinatorially diffuse' for CDCL clause-learning and that closing it needs a NEW idea — a Fourier-analytic/additive-combinatorial upper bound, a degree-4/6 SDP relaxation, or Lean-4 proof search — NOT more of the compute we have. Our ~100-core farm is strictly smaller than the cluster that already stalled. This is the R(5,5)-witness pattern, but worse: the exact tools we'd use have already been run at larger-than-our scale and hit a structural wall. The witness side (a 44-set) almost certainly doesn't exist, so the record can only move via nonexistence certification, which is the resistant pocket.","fit_score":3,"impact_score":2,"impact_reason":"Prize already claimed (Szemerédi), so $0 in play. A single new exact term r_3(212) — or a k≥4 term — is a legitimate but niche OEIS/additive-combinatorics contribution with no real-world relevance and modest field significance. High-visibility area, but the deliverable is one table entry.","verifier":"Poly-time exact AP-freeness check: for candidate S⊆[1,N], confirm no a8k core-hrs with no expectation of breaking a wall he already characterized as compute-invariant.","our_edge":"None beyond generic compute — and negative in practice: our ~100-core farm is smaller than the SLURM cluster (~7,850 worker-hrs) that already failed on the exact frontier with the exact tools. The sole tenuous edge is our LP/SDP-certificate lane for the SOS upper-bound angle the author suggests, but that is an unproven analytic research bet, not our verifier-first witness-or-nothing machinery.","verdict":"PARTIAL / WALL. Classification is PARTIAL: the headline (r_k=o(N)) is a proved asymptotic theorem (NONE for finite computation, prize already awarded), but extending the exact r_k(N) tables (OEIS A003002/3/4/5) is a genuine in-class finite sub-problem with a trivial verifier. However, beatability is a WALL: the flagship frontier r_3(212) resisted a ~7,850-worker-hour multi-paradigm campaign (Ergezer 2026) using CP-SAT+MIP+SAT — the identical arsenal we'd deploy, on a cluster larger than ours — with a flat LP relaxation and a paradigm-invariant hard pocket the author says needs new theory (Fourier/SDP/Lean), not more compute. Same shape as R(5,5)'s witness wall, only the tools we'd use have already been run past our scale and stalled. Do not attack for a record. If any effort at all, the only defensible angle is the low-impact k≥4 table terms (smaller frontier N, less-studied) or the speculative SOS/SDP upper-bound lane — both research bets, not farm wins.","lane":"exact-backtracking","board_class":"HEAVY","board_class_reason":"movable content is the exact-table certificate side (exhaustion receipts for r_k, k >= 4); the r_3(212) witness object is carried by #140","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/139","links":{"oeis":["A003002","A003003"],"arxiv":["2606.04016"]},"frontier":{"summary":"exact r_k(N) tables beyond r_3: certificate-tier (exhaustion receipts); r_3 frontier object carried by #140"}},{"id":140,"title":"r_3(N) — 3-AP-free set records (A003002)","prize":null,"statement":"Prove r_3(N) << N/(log N)^C for every C>0, where r_3(N) is the size of the largest subset of {1,...,N} with no non-trivial 3-term arithmetic progression.","finite_object":"Largest 3-AP-free subset of [1,N] (exact r_3(N) values, OEIS A003002); equivalently a max independent set in the \"3-AP\" 3-uniform hypergraph, or an UNSAT nonexistence certificate for a (K+1)-set.","current_record":"Headline PROVED asymptotically by Kelley-Meka 2023 (prize claimed). Finite table: exact r_3(N) certified through N=211 (r_3(211)=43), OEIS A003002 b-file (Cariboni). Frontier N=212 attacked 2026 by M. Ergezer, arXiv 2606.04016: 43-set witness found (lower bound), but no-44 upper bound NOT certified — 2 structurally hard chunks resisted CP-SAT/HiGHS-MIP/CDCL, dual bounds pinned at 0.0.","beatable":"WALL","beatable_reason":"Two independent walls. (1) Witness side: a 44-set in [1,212] would extend the table upward, but Ergezer 2026 searched millions of CP-SAT subproblems and found none — strong evidence it does not exist. (2) Nonexistence side: certifying r_3(212)=43 requires proving no 44-set exists, and that was JUST attacked with the identical SAT/MIP/DRAT pipeline we would use and hit a documented structural hard pocket (dual bounds at 0.0, 2 chunks pushed to Lean formal search). This is R(5,5) redux: the exact-record frontier was set in 2026 by our own toolkit and sits at the feasibility wall. Closing the 2 remaining chunks would extend the OEIS table by exactly one value (N=212) — a marginal, contested contribution, not a record we can cleanly take.","fit_score":8,"impact_score":3,"impact_reason":"Prize ($500) already claimed by Kelley-Meka 2023; the finite table extension does NOT claim it. Field value of pushing A003002 from N=211 to N=212 is low-to-moderate (one term), real-world relevance minimal. Erdős-name cachet is the only lift.","verifier":"Trivial exact check: given A ⊆ [1,N], verify |A|=K and no aCn^{2−1/r} edges (an infinite construction + asymptotic lower bound); a proof requires an asymptotic upper bound over an infinite graph class. Neither is a finite witness.","current_record":"Open. Best partial results: Füredi (1991) and Alon–Krivelevich–Sudakov (2003, dependent random choice) proved the one-sided special case — H bipartite with max degree ≤ r on one side ⇒ ex(n;H)=O(n^{2−1/r}). For general r-degenerate bipartite H the best known is the weaker AKS bound ex(n;H)=O(n^{2−1/(4r)}). No 2023–2026 resolution of the full conjecture found.","beatable":"WALL","beatable_reason":"Pure asymptotic/analytic conjecture. Progress means improving the exponent bound (a theorem) over an infinite class; there is no record number to push, no finite object beating a threshold, and no exact verifier for ≪. This is the exponent-improvement analogue of an upper-bound wall (like R(5,5)'s upper bound = NONE), with no witness side to attack.","fit_score":1,"impact_score":5,"impact_reason":"Central, named conjecture in degenerate extremal graph theory (Erdős–Simonovits), but modest $500 prize and essentially zero real-world relevance; would be a significant theoretical result if resolved.","verifier":"No poly-time exact verifier for the ≪ claim; the only checkable object (exact ex(n;H₀) at small n via ILP/SAT, cost exponential in n) is disconnected from the conjecture.","attack":"No viable finite-computation attack. Small exact ex(n;H) via ILP/SAT is feasible but does not bear on the asymptotic exponent (the superlinear regime lies far beyond computable n), so it yields no contribution to #146. Any real move is a dependent-random-choice / algebraic-construction theorem — not our farm's shape.","our_edge":"none beyond generic compute","verdict":"NONE — downgrade from the PARTIAL triage. The headline is a universal asymptotic upper bound with a hidden constant; no finite witness, no exact verifier for ≪, and the proposed \"small exact Turán numbers\" sub-problem does not inform the asymptotic exponent, so it is not genuine in-class progress on this conjecture. A new theorem (exponent improvement via DRC/algebraic constructions) is the only path. Zero fit for our verifier-first CPU farm.","lane":"wall","board_class":"NONE","board_class_reason":"no poly-time exact verifier for the asymptotic claim; small-n ex(n;H0) values are disconnected from the conjecture","wall_reason":"no poly-time exact verifier for the asymptotic claim; small-n ex(n;H0) values are disconnected from the conjecture","erdos_url":"https://www.erdosproblems.com/146","links":{"oeis":[],"arxiv":[]}},{"id":159,"title":"R(C4,Kn) sub-quadratic exponent","prize":null,"statement":"Erdős #159 ($100): there exists c>0 with R(C_4, K_n) << n^{2-c}, i.e. the Ramsey number of the 4-cycle vs the complete graph K_n admits a power-saving below exponent 2.","finite_object":"Finite sub-problem only: exact small values R(C_4,K_n). Witness = a C_4-free graph on N vertices with independence number <= n-1 (giving R > N); nonexistence = DRAT UNSAT that no such graph on N vertices exists. The headline conjecture itself has NO finite witness.","current_record":"Exact table (Radziszowski dynamic survey 'Small Ramsey Numbers'): R(C_4,K_9)=30, R(C_4,K_10)=36 (Lange, Radziszowski, Xu, 2014/2016, SAT + structured C_4-free graph generation). Next open: 39 <= R(C_4,K_11) <= 44. Asymptotic: Omega((n/log n)^{3/2}) (Spencer 1977 / projective-plane polarity graphs) <= R(C_4,K_n) <= n^{2-o(1)} type (Szemeredi; Caro-Li-Rousseau-Zhang 2000). Exponent-2 upper bound has only polylog saving; no power c>0 known.","beatable":"WALL","beatable_reason":"The $100 headline (power-saving exponent) is purely asymptotic — the Erdős-problems page explicitly states it cannot be resolved by finite computation; it needs a new theorem (NONE). The one genuine finite sub-problem — extending the exact R(C_4,K_n) table to n=11 — is a WALL: the current record (n=10, 2014) was set by Radziszowski's own RIT group using exactly our tool class (SAT + structured C_4-free graph exhaustion), and n=11 has resisted their dedicated pipeline for ~12 years. Same R(5,5)-witness situation: the record-holder already uses SAT/exhaustion, the next size hits combinatorial explosion (C_4-free graphs on 40+ vertices with the required degree structure blow up), and a 100-core farm gives no edge over a purpose-built extension pipeline. The cheap verifier does not make the search feasible.","fit_score":6,"impact_score":3,"impact_reason":"Prize is only $100 and attaches to the asymptotic exponent, which finite computation cannot claim. The reachable deliverable (an exact R(C_4,K_11) value or tightened 39-44 bounds) is a legitimate but modest entry in the small-Ramsey-numbers survey — of interest to the extremal-combinatorics niche, no prize credit, no real-world relevance.","verifier":"For exact small values: (a) verify C_4-freeness in O(N^3) (every vertex pair has <=1 common neighbor); (b) verify max independent set <= k by exhaustive/branch-and-bound MIS on N<=~44 vertices (cheap). Both poly/trivial per candidate. For the asymptotic headline: no finite verifier exists.","attack":"If attempted (not recommended): pin R(C_4,K_11). Lower half — local-search / SA + SAT to find a C_4-free graph on N=38..43 vertices with alpha<=10, encoded as SAT with C_4-free clauses (no two vertices share 2 neighbors) + independence-number cardinality constraints; verify via exact MIS. Upper half — DRAT-certified UNSAT that no (C_4,K_11;N)-graph exists, using degree-reduction lemmas (min-degree bounds from R(C_4,K_10)=36) to bound the branching, mirroring Lange-Radziszowski-Xu's one-vertex-extension exhaustion. Rough compute: the structured search space at N~40 is far larger than the K_10 case that itself took a specialized 2014 campaign; a 100-core-weeks brute attempt would very likely stall short of a decisive UNSAT — this is why it is a wall, not a farm target.","our_edge":"None beyond generic compute. The record was set by the field's leading computational Ramsey group with the same SAT+exhaustion machinery we would deploy; we have no structural lever (no symmetry-reduction or verifier advantage) that RIT's pipeline lacks.","verdict":"PARTIAL. Headline (n^{2-c} power saving) is asymptotic and explicitly not finite-computable — a new theorem is the only path (NONE for the prize). But there is a real in-class finite sub-problem: exact R(C_4,K_n) values (table exact to n=10=36; n=11 open at 39-44), a clean witness-or-UNSAT target with a trivial verifier. That sub-problem is a WALL for us — its record was set by the same SAT/exhaustion tools we'd use and has resisted the domain's dedicated group for ~12 years, with no edge on our side. Fits our machinery cleanly on paper (fit 6) but is not beatable in practice; low impact (3). Do not attack.","lane":"witness-local-search","board_class":"HEAVY","board_class_reason":"C4-freeness is cheap, but alpha <= 10 requires a measured and a-priori-bounded exact MIS adjudication; the survey does not establish the <=1 s R1 contract","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/159","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"39 <= R(C4,K11) <= 44 (Lange-Radziszowski-Xu 2014/2016)"}},{"id":161,"title":"Continuity/jumps of hypergraph discrepancy function F^{(t)}(n,α)","prize":"$500","statement":"F^{(t)}(n,α) is the smallest m such that the edges of the complete t-uniform hypergraph on n vertices can be 2-colored so that every X⊆[n] with |X|≥m contains ≥α·C(|X|,t) t-subsets of EACH color. Question ($500): for fixed n,t, as α goes from 0 to 1/2, does F^{(t)}(n,α) increase continuously or with jumps — and is there only one jump (at α=0)?","finite_object":"For fixed small n,t and rational α, F^{(t)}(n,α) is an exact finite quantity: a min over 2-colorings (for t=2, over graphs on n vertices up to iso) of the largest 'unbalanced' subset size + 1. Exhaustive-up-to-iso is feasible to ~n=11–12 for t=2 (12,346 graphs at n=8, ~12M at n=10).","current_record":"No numeric record exists — the frontier is a growth RATE, not a value. Best known: Erdős–Hajnal bounds giving F^{(t)}(n,α) growing from an iterated logarithm (small α) to a power of log n (α→1/2), the upper regime conditional on the Erdős–Hajnal–Rado conjecture. The continuity/jump question is fully OPEN; Erdős conjectured the only jump is at α=0. No 2023–2026 resolution found.","beatable":"WALL","beatable_reason":"The $500 frontier is asymptotic (growth-order of F as n→∞ across a continuum of α) — no finite witness, no record number, only a new theorem resolves it; the problem's own metadata says resolvable_by_finite_computation=False. The genuine in-class finite sub-problem (exact small-case F-tables) is computationally trivial but doubly non-illuminating: (1) the phenomenon is iterated-logarithmic, so log*n / log n / (log n)^c are indistinguishable order-1 constants at any reachable n (≤~12); (2) at finite n, F(n,α) is trivially a step function in α (jumps at every achievable rational density j/C(k,2)) — an integrality artifact unrelated to the asymptotic growth-order jump the conjecture is about. So the finite table would mislead, not help, and there is no established record it beats.","fit_score":3,"impact_score":4,"impact_reason":"$500 prize (small). Field significance moderate: classical Erdős discrepancy/Ramsey theory, tied to tower-type Erdős–Hajnal–Rado growth questions. Real-world relevance nil. But entirely out of reach for our method, so realizable impact for us is ~0.","verifier":"Poly-time exact checker: for a given coloring, over all 2^n subsets X compute minority-color density min(e_red,e_blue)/C(|X|,t); the condition fails at m iff some |X|≥m has density<α. O(2^n · poly) per coloring — trivial for small n. Verifier is clean; the object it verifies is just worthless toward the prize.","attack":"MOVABLE only in the vacuous sense: exhaustive-up-to-iso enumeration of graphs (nauty/geng) for t=2, n≤11–12, and t=3 for tiny n, computing α↦F^{(t)}(n,α) step functions via a 2^n-subset density scan per coloring. ~hours on a few cores; no SAT/DRAT needed. But this produces a table (absent from OEIS) that cannot see the iterated-log regimes and whose 'jumps' are integrality artifacts — it does not touch the $500 question. Not worth farm time.","our_edge":"None beyond generic compute. Our verifier-first exact-witness machinery maps mechanically onto the small-case checker, but the prize is an asymptotic growth-order theorem where witnesses/records don't exist. Unlike R(5,5) (where the lower-bound witness was a concrete finite graph) or kissing (a concrete config beating a number), there is no finite object here whose discovery constitutes progress on the stated question.","verdict":"PARTIAL-in-class but WALL-for-us. There IS a genuine finite sub-problem — exact 2-coloring search over the complete t-uniform hypergraph with a poly-time checker to tabulate small-n,t exact values of F^{(t)}(n,α) — so it is not NONE. But the $500 headline is asymptotic (growth-order continuity/jumps as n→∞) and explicitly not finite-computable; the finite sub-problem is both computationally trivial AND non-illuminating (iterated-log scale invisible at reachable n; finite 'jumps' are integrality artifacts distinct from the asymptotic jump). There is no numeric record to beat. Do not attack — no farm advantage, no prize-relevant witness. Class: PARTIAL; beatability: WALL.","lane":"wall","board_class":"NONE","board_class_reason":"no numeric record exists - the frontier is a growth rate, not a value","wall_reason":"no numeric record exists - the frontier is a growth rate, not a value","erdos_url":"https://www.erdosproblems.com/161","links":{"oeis":[],"arxiv":[]}},{"id":165,"title":"Asymptotic formula for R(3,k)","prize":"$250 (for the asymptotic formula only; no prize attaches to any finite exact-value contribution)","statement":"Give an asymptotic formula for the Ramsey number R(3,k) (largest n for which a triangle-free graph on n vertices with independence number < k exists, plus one). Erdős #165, $250 prize.","finite_object":"Extremal triangle-free graph on n vertices with independence number <= k-1 (an R(3,k,n)-graph); or a SAT-certified nonexistence proof at a given (k,n).","current_record":"Headline asymptotic: R(3,k)=Theta(k^2/log k) — lower (1/4-o(1))k^2/log k by Shearer/Bohman-Keevash, refined by Campos-Jenssen-Michelen-Sahasrabudhe (arXiv 2505.13371, 2025); upper by Bohman-Keevash. Finite table: R(3,9)=36 (McKay-Zhang, ~1990s, last exact value); R(3,10) in {40,41} — lower 40 = Exoo 39-vertex construction (simulated annealing/cyclic), upper 41 = Angeltveit 2024 (arXiv 2401.00392, custom gluing, ~3 CPU-years, no DRAT); R(3,11) in {47,50}; R(3,12-16) wider gaps.","beatable":"WALL","beatable_reason":"The $250 prize is purely asymptotic/analytic — the page explicitly says it cannot be resolved by finite computation; no witness route to the prize. The genuine finite sub-problem (closing R(3,10), improving R(3,11+) lower bounds) is also a wall for our farm: (a) the lower-bound witness record was set by simulated-annealing/Metropolis — the exact tool we would deploy — and has been stuck at a 39-vertex graph for years, strongly implying R(3,10)=40 (better witness does not exist); (b) the nonexistence/SAT direction is defeated by clause explosion (the no-independent-10-set constraint is ~C(40,10)=8.5e8 width-45 clauses, infeasible to even instantiate), and Angeltveit states the orderly-generation route needs orders of magnitude beyond 3 CPU-years; (c) specialized nauty-based gluing codebases (McKay/Angeltveit/Radziszowski/Goedgebeur) dominate generic solvers, and SAT experts (Heule) have cracked adjacent problems but not R(3,10). This is the R(5,5)-witness situation exactly: computationally-shaped but past the feasibility horizon, record held by the same tools we'd bring.","fit_score":6,"impact_score":6,"impact_reason":"Settling R(3,10) exactly would be the first new exact R(3,k) value in ~30 years and a genuine headline in the small-Ramsey community — real field significance. But the $250 prize is unreachable (asymptotic-only), there is no real-world relevance, and the finite contribution would be non-prize-claiming.","verifier":"Witness (lower bound): O(n^3) triangle-free check + independence-number check alpha<=k-1 via a max-independent-set solver or C(n,k) subset enumeration (C(40,10)~8.5e8, feasible once). Nonexistence (upper bound): DRAT certificate from a SAT UNSAT run — clean in principle but the CNF encoding is intractable to build for k=10.","attack":"Two finite routes, both wall: (1) MIS/SAT/simulated-annealing re-search for a 40-vertex triangle-free graph with alpha<=9 -> would give R(3,10)=41; SA farm cost days-weeks but Exoo has already exhausted this with the same method. (2) Cube-and-conquer SAT with heavy symmetry breaking for nonexistence -> R(3,10)=40 with DRAT; but the independence constraint yields ~8.5e8 clauses, and the enumeration route is orders of magnitude beyond the 3 CPU-years already spent. Neither is within reach of a ~100-core farm.","our_edge":"none beyond generic compute — Exoo already uses simulated annealing (our lower-bound tool), specialized orderly-generation/gluing code dominates the upper-bound side, and the SAT encoding is beyond current feasibility for everyone.","verdict":"PARTIAL classification confirmed but BEATABLE=WALL. Headline is asymptotic/analytic (NONE-type, prize unreachable by compute). The in-class finite sub-problem — exact R(3,10)/R(3,11) values and better constructions — is a textbook match to our R(5,5) witness-or-nonexistence farm in SHAPE, but the actual instances are past the feasibility horizon: the lower-bound witness record is held by the very SA/Metropolis tool we'd use (and appears exhausted, suggesting R(3,10)=40), while the nonexistence/SAT direction dies on clause explosion and is dominated by decades of specialized nauty gluing code. Do not attack — no movable frontier and no claimable prize. Same disposition as the R(5,5) witness: computationally-shaped, at the wall.","lane":"witness-local-search","board_class":"HEAVY","board_class_reason":"lower-bound witness needs an independence-number <= 9 verification: ~8.5e8 subset scan / MIS run per candidate - minutes, not seconds (R1 fails)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/165","links":{"oeis":[],"arxiv":["2401.00392","2505.13371"]},"frontier":{"summary":"R(3,10) in [40,41] (R(3,9)=36, McKay-Zhang ~1990s)"}},{"id":166,"title":"Erdős #166 — R(4,k) ≫ k³/polylog","prize":"$250 (already settled — headline PROVED by Mattheus–Verstraete 2023)","statement":"Prove R(4,k) ≫ k³/(log k)^{O(1)}. Headline PROVED by Mattheus–Verstraete 2023 (R(4,k) ≫ k³/(log k)⁴), matching the Ajtai–Komlós–Szemerédi upper bound up to polylog. The finite-computation sub-problem in-class: the exact small off-diagonal values R(4,k) — R(4,5)=25, R(4,6)∈[36,40], R(4,7)∈[49,58], R(4,8)∈[59,79] — which are R(5,5)-style witness/nonexistence computations with an exact edge-coloring verifier.","finite_object":"A 2-coloring of K_n with no red K₄ and no blue K_k (equivalently a K₄-free graph on n vertices with independence number < k) — a \"(4,k)-Ramsey graph.\" Lower bound = exhibit one on n vertices; upper bound = certify none exists on n vertices.","current_record":"Headline: R(4,k) ≫ k³/(log k)⁴ — Mattheus & Verstraete, Annals 2023 (PROVED; asymptotic; resolves #166, $250 prize already settled). Small values: R(4,5)=25 (McKay–Radziszowski 1995). R(4,6): lower 36 = Exoo 2012 (cyclic/computer search, Electron. J. Combin.); upper 40 = Angeltveit & McKay 2019 (isomorph-free orderly generation, per Radziszowski \"Small Ramsey Numbers\" survey rev. 2024-09-06). R(4,8) lower 58 = Exoo 2012 (was 56). SAT solvers now standard here (e.g. Wesley, arXiv 2509.03784, 2025, multicolor).","beatable":"WALL","beatable_reason":"Three walls. (1) The headline is PROVED and asymptotic — no finite object, no claimable prize; pure NONE. (2) The finite sub-problem is genuinely in-class but is the textbook \"computational-looking record already at the wall,\" exactly like the R(5,5) witness was for us. The specialized community owns strictly better tooling than our generic farm: Exoo's targeted cyclic/quasi-cyclic constructions for lower bounds and McKay's nauty-based isomorph-free orderly generation for upper bounds — and R(4,6)'s lower bound has been frozen at 36 since 2012 despite exactly this machinery hammering it. (3) The tools we'd bring (SAT + DRAT, local search, MIS) are the same tools the frontier is already running (SAT is now standard in this niche). \"Record set by the same-or-better tools we'd use\" = no edge. A 100-core farm would reproduce known values, not beat them. Lower-bound ticks on R(4,7)/R(4,8) are marginally conceivable but low-probability on Exoo's home turf and carry no prize — a non-claiming ledger entry at best.","fit_score":7,"impact_score":4,"impact_reason":"Headline is a famous $250 Erdős prize but already resolved (Mattheus–Verstraete) — nothing to claim. Fully determining R(4,6) would be a genuinely celebrated result (one of the most-wanted small Ramsey numbers), high field significance — but it is NOT what #166 asks, carries no prize, and is unreachable by us. A realistically achievable outcome (a single lower-bound tick on R(4,7)/R(4,8)) is minor and non-prize. Net achievable-for-us impact is low.","verifier":"Exact and cheap: given a candidate graph on n≤~80 vertices, check no K₄ (O(n⁴) clique enumeration) and no independent set of size k (bounded clique search on complement; trivial at these n). Nonexistence at a size via SAT + DRAT certificate over the C(n,2) edge variables with clique/independent-set clauses — the same encoding used for R(5,5). Verifier cost negligible; the search is the cost.","attack":"If attempted anyway (not advised): (a) lower bound — SA / tabu local search plus circulant/Cayley-graph seeding over C(n,2) edges of K₃₆–K₃₉ maximizing \"(4,6)-feasibility,\" restart farm ~100 cores × weeks; (b) nonexistence — CNF encode \"K_n has red K₄ or blue K₆\" and hand to CaDiCaL/Kissat with symmetry breaking, emit DRAT. Both are exactly what Exoo/McKay/SAT-community already do at greater depth; expected outcome is rediscovery of 36 and no DRAT closure below 40.","our_edge":"None beyond generic compute. Our R(5,5) and kissing precedents are the same verifier-first pattern, but here the incumbents (Exoo cyclic search, McKay orderly generation, active SAT groups) have deeper domain-specific search than our farm — the opposite of a green-field board.","verdict":"PARTIAL problem, but WALL for us — DO NOT ATTACK. Headline is PROVED/asymptotic (NONE). The in-class finite sub-problem (exact small R(4,k)) maps cleanly to our SAT+DRAT / witness-search farm (why fit is high), but it is the R(5,5) situation over again: the record is held by specialists with strictly better tooling than our generic compute and has been static for 12+ years, and our candidate tools (SAT/DRAT/local search) are precisely the incumbent tools. No edge, no claimable prize. File under \"computational-looking but at the wall,\" same disposition as the R(5,5) witness hunt.","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = 2-coloring on <= ~40 vertices; K4/K6 clique scans trivial; R(4,6) >= 37 witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/166","links":{"oeis":[],"arxiv":["2509.03784"]},"frontier":{"summary":"R(4,6) in [36,40] (lower 36 = Exoo 2012)"}},{"id":182,"title":"Max edges without a k-regular subgraph (Erdős–Sauer)","prize":"$100 (offered by Erdős for k=3) — effectively resolved; not collectible via finite computation.","statement":"Let k≥3. What is the maximum number of edges f_k(n) that an n-vertex graph can have without containing a k-regular subgraph? Erdős–Sauer asked (with a $100 prize for k=3) whether it is n^{1+o(1)}.","finite_object":"A finite object exists only in the (untracked) sub-problem: an n-vertex extremal graph with f_k(n) edges and no k-regular subgraph, plus a certificate that every graph with f_k(n)+1 edges contains one. No such small-n table is published or maintained.","current_record":"Headline PROVED: f_k(n)=Θ_k(n log log n). Upper bound O(n log log n) with C(k)≪k^2 by Chakraborti–Janzer–Methuku–Montgomery (arXiv 2411.11785, 2024), following Janzer–Sudakov's resolution (Forum Math Pi 11 e19, 2023) that settled the $100 k=3 case; matching lower-bound construction by Pyber–Rödl–Szemerédi (1995). No published or OEIS small-n exact table for f_k(n).","beatable":"WALL","beatable_reason":"The asymptotic answer is a THEOREM (Janzer–Sudakov 2023, tightened 2024) — no computation touches it and the $100 prize is effectively resolved. The only finite angle is exact small-n f_k(n), but (1) there is NO record to beat: no OEIS entry, no published small-n table, no community-tracked frontier — we would be minting an uncited curiosity, not moving a recognized number; (2) the verifier is NP-complete, so this fails the STRONG poly-time-verifier bar; (3) fixing f_k(n) is a two-level (Σ₂-shaped) search — construct a no-k-regular graph AND prove every (m+1)-edge graph forces one — requiring canonical-graph exhaustion (nauty) that blows up past n≈10–12. Unlike R(5,5)/kissing, there is no clean single witness-or-DRAT target and no held record whose defeat would count.","fit_score":3,"impact_score":2,"impact_reason":"The $100 prize question is PROVED (asymptotically tight, 2023–2024); no open prize remains. Exact small-n values are not a tracked target (no OEIS, no published table), have no downstream or real-world relevance, and would be an uncited exercise. Field significance of any small computation here is negligible.","verifier":"\"Contains a k-regular subgraph?\" is NP-COMPLETE for every k≥3 (Chvátal/Garey–Johnson; cubic-subgraph NP-c even on bipartite max-deg-4). So there is NO poly-time exact verifier. Existence of a k-regular subgraph has a trivial poly certificate (exhibit it); NON-existence needs a SAT-UNSAT/DRAT proof. Determining f_k(n) additionally requires the ∀-over-all-graphs upper bound (a second, harder exhaustion). Cost: per-graph SAT is cheap for n≲12 but the outer max-edge + all-graph search explodes.","attack":"If pursued anyway (low value): CEGAR loop — outer ILP/SAT maximizes edge count on n labeled vertices; inner oracle SAT-checks for a k-regular subgraph and, when found, adds a blocking/forbidding clause; iterate to the max-edge no-k-regular graph. Pair with nauty canonical enumeration to certify the +1-edge upper bound. Feasible only to n≈10–12 on a ~100-core farm (hours–days); DRAT-certifiable per instance but no external record to validate against. Beyond n≈12 the labeled-graph search and NP-complete inner checks become infeasible.","our_edge":"None beyond generic compute. Our SAT+DRAT nonexistence machinery fits the \"no k-regular subgraph\" leg, but there is no target record, the prize is claimed, and the upper-bound exhaustion is outside our verifier-first witness-or-nothing sweet spot.","verdict":"PARTIAL problem-class, WALL for our farm — PASS. The headline Erdős–Sauer question is a proved theorem (Θ_k(n log log n), Janzer–Sudakov 2023 / CJMM 2024), immovable by computation. The one finite sub-problem — exact small-n f_k(n) — is well-defined and in-class but has no published record/OEIS table to beat, an NP-complete verifier (fails the poly-time-verifier bar), and a doubly-quantified graph-exhaustion that explodes past n≈12. There is no witness-or-DRAT target whose defeat would register as progress. Recommend no attack; impact and fit are both low.","lane":"wall","board_class":"NONE","board_class_reason":"'contains a k-regular subgraph' is NP-complete for k >= 3: no poly-time exact verifier","wall_reason":"'contains a k-regular subgraph' is NP-complete for k >= 3: no poly-time exact verifier","erdos_url":"https://www.erdosproblems.com/182","links":{"oeis":[],"arxiv":["2411.11785"]}},{"id":183,"title":"Limit of multicolor triangle Ramsey R(3;k)^{1/k}","prize":"$250 (Erdős)","statement":"Determine lim_{k→∞} R(3;k)^{1/k}, where R(3;k)=R_k(K_3) is the least n such that every k-coloring of K_n's edges yields a monochromatic triangle.","finite_object":"A k-edge-coloring of K_n with no monochromatic triangle (lower-bound witness R_k(3)>n), or a DRAT proof that none exists at n (upper bound). Both are finite; the headline limit is not.","current_record":"Exact: R_2=6, R_3(3)=17 (Greenwood–Gleason 1955). Bounds (Radziszowski DS1.18, 2026 / OEIS A003323): R_4(3)∈[51,62] (51=Chung 1973, proven un-improvable by natural constructions [Ana1,Ana2]; 62=Fettes–Kramer–Radziszowski); R_5(3)∈[162,307] (Exoo); R_6(3)∈[538,1838] (Fredricksen–Sweet); R_7(3)∈[1698,12861] (Rowley); R_8(3)≥52888, R_9(3)≥178059 (AgCP+, 2024); k=10–15 improved 2024 via generalized Schur-number templates. Limit itself known only to lie in [~3.2, ∞]; finiteness OPEN.","beatable":"WALL","beatable_reason":"Headline is asymptotic and needs a theorem (prove the limit finite-with-value, or prove it infinite) — no finite witness resolves it. The genuine finite sub-frontier (small-case R_k(3) bounds, OEIS extension) is either provably stuck (R_4(3)=51 un-improvable by the natural construction classes; the 51–62 gap SAT-infeasible, far beyond R(5,5) which was already at the witness wall) or is actively moving only in the hands of specialists (Exoo, Rowley, Schur-template groups; 2024 AgCP+) using exactly the SAT / cyclic / local-search / Schur-heuristic tools we would deploy. No edge beyond generic compute; no single R_k(3) improvement claims the prize.","fit_score":5,"impact_score":5,"impact_reason":"Famous named Erdős problem, deep and well-known, but $250 is modest and any finite contribution (an R_k(3) construction) is incremental and non-prize-claiming; real-world relevance low.","verifier":"Per-color triangle-freeness of an n-vertex k-edge-coloring: O(k·n^3) exact (or O(k·n^ω)). Nonexistence side = SAT/DRAT over the coloring CNF, infeasible at these n.","attack":"If attempted: CNF-encode 'k-color edge-coloring of K_n, each color triangle-free' (Schur-number style), attack with cyclic-coloring exhaustion + Schur-template local search / simulated annealing on ~100 cores to try to beat e.g. R_5(3)≥162, R_6(3)≥538, or extend A003323 at k=10–15; verifier O(k·n^3). Realistic compute: weeks of 100-core with ~no expected improvement — this is precisely the search Exoo/Rowley ran for years without a new template idea.","our_edge":"None beyond generic compute. The movable sub-frontier is occupied by the exact toolkit we'd use (Exoo cyclic search, Rowley/Schur linear templates, Codish/Wesley SAT), tuned over decades.","verdict":"PARTIAL structurally but WALL in practice — PASS. The witness/verifier shape maps cleanly onto our verifier-first farm (trivial O(k·n^3) triangle check), but the prize is asymptotic (theorem-only) and every movable finite record is either proven-stuck (R_4(3)=51), SAT-infeasible to close (upper bounds), or already held by specialists using our own tools with no edge for us. Same lesson as R(5,5): a computational-looking record already at the wall.","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = k-edge-coloring; per-color triangle scan O(k n^3), exact; R_4(3) >= 52 witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/183","links":{"oeis":["A003323"],"arxiv":[]},"frontier":{"summary":"R_4(3) in [51,62] (Chung 1973 lower), R_5(3) in [162,307] (Exoo)"}},{"id":211,"title":"Erdős #211 — Beck/Szemerédi-Trotter (finite residue: orchard 3-point lines)","prize":"$100 (Erdős #211, but problem is SOLVED — not claimable). Orchard sub-problem carries no formal prize.","statement":"Let 1≤k1/3). Both are ASYMPTOTIC-construction prizes — a finite witness cannot claim either.","statement":"Let f(N) be the max size of A⊆{1..N} such that all sums a+b+c (a,b,c∈A) are distinct up to permutation (a B_3 / B_3^+ set). Erdős (via Bose) asks: is f(N)~N^{1/3}? OPEN, $100.","finite_object":"Maximum-cardinality B_3 set in {1..n} (all multiset triple-sums distinct) = the exact value f(n) tabulated in OEIS A387704.","current_record":"FINITE frontier: OEIS A387704 b-file, n=0..150, a(150)=8 — Sharvil Kesarwani, Dec 2025, via straightforward branch-and-bound backtracking (single machine). ASYMPTOTIC frontier (the actual conjecture): f(N)/N^{1/3} pinned in [~1, ~2.62]; lower (1+o(1))N^{1/3} = Bose–Chowla construction (1962/1960s), upper ~ (18N)^{1/3} counting bound with constant improvements (Ruzsa and others); conjecture says the constant →1. No one has moved the asymptotic in decades.","beatable":"MOVABLE","beatable_reason":"FRONTIER CORRECTED 2026-07-16: the finite table sub-problem is not an independent target — A387704 is A227358 in disguise (translation invariance), so 'extending the table' needs no search through n=309, and beyond that it coincides with #41's open a(11). Do not route compute here; route it to #41. The asymptotic conjecture f(N)~N^{1/3} remains a WALL as before.","fit_score":7,"impact_score":3,"impact_reason":"Famous Erdős/Bose–Chowla problem, but the only finite-computable piece (extending the A387704 exact table) contributes ~nothing to the actual conjecture and claims no prize — the sequence grows so slowly that even n~1000 stays near value 10, orders of magnitude below any asymptotic signal. Value is a modest OEIS b-file extension that the next better solver supersedes. Prize weight for us is effectively 0 (asymptotic, unclaimable).","verifier":"Given a candidate set A of size k: enumerate all C(k+2,3)=(k+2)(k+1)k/6 multiset triple-sums into a bitset over [3,3n] and check all distinct — O(k^3) exact, trivially cheap. Certifying f(n)=v (nonexistence of a size-(v+1) set in {1..n}) requires an EXHAUSTIVE branch-and-bound tree (DRAT-style search certificate), the expensive half.","attack":"Do not attack the table — it is determined through n=309 by the identity A387704(n)=max{k: A227358(k)<=n-1} (proof: translation invariance; verified on all published terms). The open cell (jump to 11) IS #41's a(11), owned by the #41 lane. A community contribution extending the OEIS b-file from 150 to 309 with this identity is the honest next act.","our_edge":"Clean methodological fit to our verifier-first exhaustion farm — same shape as Sidon/kissing record hunts (exact O(k^3) checker + branch-and-bound witness/nonexistence). Concrete edge over the current holder: they used a single-machine Mathematica backtracker to n=150; a C++ B&B (or CP-SAT) on ~100 cores is a straightforward multi-order speedup. But the edge only extends a low-value table, not the frontier that matters.","verdict":"PARTIAL — Headline f(N)~N^{1/3} is asymptotic (NONE for the prize; a hard theorem-only wall). BUT there is a real in-class finite sub-problem: the exact small-case table OEIS A387704 currently stops at n=150 (a(150)=8), computed by a simple single-machine backtracker in Dec 2025. Our branch-and-bound + exact O(k^3) verifier + CPU farm can legitimately extend that table and pin the first jumps to 9/10 (MOVABLE). This is a genuine but low-impact, non-prize-claiming contribution — the extended values do not bear on the asymptotic conjecture. Recommend: skip as a flagship target; at most a cheap opportunistic b-file extension if a B_3/Sidon solver already exists in-house.","lane":"exact-backtracking","board_class":"READY","board_class_reason":"witness = subset of {1..N}; O(k^3) triple-sum distinctness, exact; first-jump frontier witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/241","links":{"oeis":["A387704"],"arxiv":[]},"campaign_finding":" 2026-07-16 CLOSED-BY-CROSSREF: A387704(n)=max{k:A227358(k)<=n-1} (translation invariance), machine-checked 0/151 mismatches; first jump to 9 = 209; table free to n=309; open content = #41 a(11). The 'hobbyist backtracker' table was A227358 in disguise — third stale-frontier catch of the day (#552 a(17), f(39), now this).","frontier":{"summary":"CLOSED BY CROSS-REFERENCE (2026-07-16): A387704(n) = max{k : A227358(k) <= n-1} (B_3 is translation-invariant, so a k-element B_3 subset of {1..n} exists iff a k-mark B_3 ruler of length <= n-1 exists). Identity machine-checked against all 151 published terms (0 mismatches). Hence the table is determined through n=309 by Tromp's exhaustive A227358 values: a(n)=8 for 130<=n<=208, FIRST JUMP TO 9 AT n=209, a(n)=9 for 209<=n<=309, jump to 10 at n=310. The only open content is the jump to 11 at A227358(11)+1 <= 446 — exactly problem #41's open cell a(11)."},"p42_slug":"b3-subset-first-jump-9"},{"id":500,"title":"Turán tetrahedron K_4^3 extremal number","prize":"$500 (Erdős — for the asymptotic density of any single K_ℓ^r, ℓ>r≥3; unclaimed)","statement":"Determine ex_3(n, K_4^3): the maximum number of triples on n vertices such that no 4 vertices span all 4 of their triples (tetrahedron-free). Turán (1961) conjectured the extremal density is 5/9.","finite_object":"A K_4^3-free 3-uniform hypergraph on n vertices with MORE than A140462(n) triples (would disprove Turán), OR a DRAT certificate that none exists with A140462(n)+1 triples (exact upper bound at that specific n).","current_record":"Finite lower bound = Turán's construction = A140462(n), a closed-form polynomial (a(3k)=5/2 k^3-3/2 k^2, etc.), believed exact; multiple extremal families by Brown, Kostochka, Fon-der-Flaass, Frohmader, Razborov. Asymptotic upper bound on the density π(K_4^3) ≤ 0.561666 (Razborov 2010, flag-algebra SDP; earlier Chung–Lu 0.5936). Conjectured density 5/9 ≈ 0.5556. NO published table of exact small-n ex_3(n,K_4^3) values exists (confirmed across searches). No movement 2023–2026 on the exact-value or asymptotic front via combinatorial search; only ℓ_2-norm variants (Balogh et al. 2022, 2025) and flag-algebra refinements.","beatable":"WALL","beatable_reason":"Three-way wall. (1) The headline $500 is the ASYMPTOTIC density (conjectured 5/9) — pure analytic, needs a theorem, no finite witness. (2) The finite lower-bound \"record\" IS a closed-form formula (A140462, Turán's construction), believed tight; beating it = disproving Turán's 1961 conjecture, astronomically unlikely at small n. (3) The finite upper-bound (prove no K_4^3-free graph on n vertices exceeds the formula) is exponentially degenerate: Frohmader proved Ω(6^{n/3}) non-isomorphic conjectured-extremal hypergraphs exist — the literature names this exact degeneracy as WHY the problem is hard, and it makes DRAT-UNSAT proofs blow up past tiny n.","fit_score":4,"impact_score":3,"impact_reason":"Famous problem, real $500 prize, high field significance in extremal combinatorics — but ALL of that attaches to the asymptotic 5/9 conjecture, which finite computation cannot touch. The only thing our farm could realistically produce (one more small-n exact value) is a curiosity with zero bearing on the prize or the frontier, which is precisely why no such table has ever been published. Realistic achievable impact ≈ nil.","verifier":"K_4^3-free check: for every one of the C(n,4) 4-subsets, at least one of its 4 triples is absent. O(n^4), exact, poly-time — an excellent cheap verifier. (Cost is trivial; it's the search/UNSAT-proof side that walls out.)","attack":"ILP: binary x_T per triple; for each 4-set, sum of its 4 triples ≤ 3; maximize sum(x). Or SAT: C(n,3) vars, C(n,4) tetrahedron clauses + a cardinality constraint ≥ formula+1, prove UNSAT with DRAT, plus S_n symmetry-breaking. Rough compute: n≤10 minutes; n=11–13 hours-to-hard even with aggressive symmetry breaking; the Ω(6^{n/3}) near-extremal degeneracy walls it beyond that. A ~100-core farm does not change the exponential, and even a success only CONFIRMS ex(n)=formula for one more n — a curiosity that (by the same degeneracy) tells you nothing about the asymptotic frontier.","our_edge":"None beyond generic compute. The one competitive frontier — the asymptotic UPPER bound — lives in SDP/flag-algebra machinery (Razborov, Baber–Talbot, Vaughan), not our SAT/ILP/MIS farm; we have no edge there. The \"extend A140462\" hook is a mirage (it's a formula, not computed data).","verdict":"SKIP. PARTIAL by the letter — a genuine in-class finite sub-problem exists (exact ex_3(n,K_4^3) via ILP/SAT with an O(n^4) exact verifier) — but beatable = WALL and impact ≈ nil. This is R(5,5) inverted: there the movable computational part was the witness/lower bound and the upper bound was NONE; here the lower bound is a tight closed formula (A140462) and only the exponentially-degenerate upper bound is computational. The triage's \"extend OEIS A140462\" premise is a miscategorization: A140462 is Turán's conjectured construction value in closed form (known to n=10000), not a table of computed extremal numbers — there is nothing to extend. The $500 is asymptotic and untouchable by our tools; the asymptotic upper-bound frontier is SDP/flag-algebra where we have no edge. Do not attack.","lane":"wall","board_class":"NONE","board_class_reason":"finite lower bound is a closed-form construction (A140462) believed exact, not an open tracked frontier; density side carried by #712","wall_reason":"finite lower bound is a closed-form construction (A140462) believed exact, not an open tracked frontier; density side carried by #712","erdos_url":"https://www.erdosproblems.com/500","links":{"oeis":["A140462"],"arxiv":[]}},{"id":548,"title":"Erdős–Sós conjecture (trees in dense graphs)","prize":"$100","statement":"Every graph on n vertices with at least (k-1)/2·n + 1 edges contains every tree on k+1 vertices (k edges); equivalently average degree > k-1 forces every k-edge tree. Open, $100, believed TRUE.","finite_object":"A single counterexample graph G plus a specific (k+1)-vertex tree T: e(G) ≥ (k-1)/2·|V(G)| + 1 yet T ⊄ G. Finite and cheaply checkable, but almost certainly nonexistent and unbounded in n/k.","current_record":"No numeric record — open conjecture. Best-known frontier: proven for large k (Ajtai–Komlós–Simonovits–Szemerédi, ~1990s announcement); large trees with near-spanning host (Reed–Stein, arXiv:2405.15733, 2024 rev. 2025); spiders (Fan–Sun, all leg lengths; Woźniak legs≤2); caterpillars (Perles, 2017); bounded-degree trees + dense hosts (1906.10219); near-diagonal k=n-1,n-2,n-3 (Woźniak), n-4 (Tiner), n=k+4 (Wang/Li 2014). No counterexample known or expected.","beatable":"WALL","beatable_reason":"Two-direction split. Positive direction (conjecture true) is NOT finitely certifiable: for fixed k the host size n is unbounded and no absolute bound reduces it to a finite check — needs a theorem, not compute. Counterexample direction is the only finite-witness path, but the object is believed not to exist: AKSS proved ES for all large k (so any counterexample is small-k), and small k is exactly where it is most verified (paths, spiders incl. k≤10, caterpillars, bounded-degree+dense hosts, n=k+O(1), Reed–Stein 2024 near-spanning). A witness hunt here is R(5,5)-witness-style: overwhelming expert belief and proof progress say the object isn't there.","fit_score":3,"impact_score":5,"impact_reason":"Famous, long-standing named conjecture (high field significance); disproof would be a major result. But only a $100 prize, near-zero realistic success probability (conjecture believed and largely proven), and no real-world relevance. Our realistically achievable contribution is null.","verifier":"Exact and cheap for a CLAIMED counterexample: edge-count check (trivial) + subgraph-isomorphism T↪G for a fixed (k+1)-vertex tree T, poly in n via color-coding / bounded-pattern search (O(n^{O(k)}) naive, FPT in k). The verifier is fine; the problem is there is no bounded search space to run it over (n unbounded) and the positive direction has no finite certificate.","attack":"Only conceivable attack: SAT/ILP or exhaustive counterexample hunt — for small trees T (small k) encode \"G on n vertices, e(G) ≥ (k-1)/2·n+1, and G avoids T\" and search over n up to feasibility (~n≤20–25, k≤~8) on the CPU farm; verify any hit with the exact subgraph-iso checker. Rough compute: modest (hours–days on ~100 cores). Expected outcome: UNSAT everywhere reachable (region already covered by AKSS + small-case theorems), producing no new result. No positive-direction certificate is possible (unbounded n). Recommend NOT pursuing.","our_edge":"None beyond generic compute. Our exact-verifier moat doesn't help: the verifier is the easy part; the hard part (bounded search domain / positive certificate) does not exist for this problem.","verdict":"PARTIAL — a genuine finite sub-problem exists (exhaustive small-(n,k) verification / bounded SAT counterexample search with a clean exact poly verifier), so not NONE; but not STRONG: no numeric record to push and the positive direction is not finitely certifiable (unbounded n). Beatable status is WALL: the only finite object (a counterexample) is nearly certainly nonexistent, ruled out at large k by AKSS and at small k by extensive theorems.","lane":"wall","board_class":"NONE","board_class_reason":"no finite search space (n unbounded) and no numeric record; positive direction has no finite certificate","wall_reason":"no finite search space (n unbounded) and no numeric record; positive direction has no finite certificate","erdos_url":"https://www.erdosproblems.com/548","links":{"oeis":[],"arxiv":["2405.15733"]}},{"id":552,"title":"R(C_4, S_n): C4-vs-star Ramsey","prize":"$100 (Erdős)","statement":"Determine the Ramsey number R(C_4, S_n) where S_n = K_{1,n} is the star on n+1 vertices. In particular: is it true that for any c>0 there are infinitely many n with R(C_4, S_n) <= n + sqrt(n) - c?","finite_object":"For fixed n, a simple red graph on m labeled vertices with no C4 and minimum degree at least m-n. Its complement then has no K1,n. Maximizing m improves the lower bound R(C4,K1,n) >= m+1.","current_record":"Exact values known in the literature for all n <= 38 (Boza arXiv:2409.12770 table with references; e.g. f(12..17)=17..22 published 1975-2018 — OEIS A006672's b-file, stopping at n=11, is far behind the survey literature). P42 certificates for a(12..16) and the a(17) lower witness are independent machine-checkable re-derivations, not novel records.","beatable":"MOVABLE","beatable_reason":"FRONTIER CORRECTED 2026-07-16: the n=17 cell was never open — R(C4,K1,17)=22 is Parsons 1975. Do NOT route compute to n=17,m=22. Remaining open cells (n=39,42,44,...) reduce to extremal C4-free constructions on 45+ vertices where ex(m;C4) is itself only bracketed (McKay/Alekseyev) — campaign-scale, actively worked field (Boza Jun 2026), high racing risk.","fit_score":7,"impact_score":3,"impact_reason":"Prize is only $100 and is unclaimable by finite computation. Classic Erdős/Parsons problem with elegant projective-plane structure (moderate field interest), but a single new OEIS table term is a minor, non-prize-claiming contribution with negligible real-world relevance. The genuinely significant part (the asymptotic conjecture) is out of computational reach entirely.","verifier":"Represent the red graph. Check C_4-freeness by requiring every vertex pair to have codegree <= 1. A blue S_n is absent exactly when every blue degree is <= n-1, equivalently every red degree is >= m-n on an m-vertex complete graph. Both checks are exact and tiny for m ~ 15-45.","attack":"The n=17 cell is CLOSED (Parsons 1975) and n=39 is CLOSED (this repository, 2026-07-16: ER_7-deletion witness, certificates/erdos-552-f39/). Remaining open cells n=42 (needs C4-free 49-vtx min-deg-7) and n=44 (51-vtx min-deg-7): polarity-deletion families ER_7/ER_8 are certified UNSAT for both — a witness needs a different construction or general SAT on the full graph space. Require the shipped codegree/degree verifier for witnesses; DRAT/LRAT for nonexistence.","our_edge":"None beyond generic compute. The verifier is trivial (no moat), the object is small, and the exact table is already being actively extended by an outside worker (Alex Towell, added a(11) 11 days ago) with the same SAT tooling we'd deploy. We cannot claim the $100 prize this way (asymptotic), and on the table we'd merely be racing.","campaign_finding":"2026-07-13: local PySAT/CaDiCaL edge-SAT passes found exact witnesses meeting Parsons' upper bound for n=12..16 and a 21-vertex witness proving R(C4,K1,17) >= 22. Dependency-free verifiers recomputed all degrees and pair-codegrees; CHRONOS independently replayed n=12 and a separate hostile checker replayed all six graphs. A limited 1,000,000-conflict pass on n=17,m=22 returned UNKNOWN, not UNSAT. Do not repeat the certified regions or treat that timeout as a bound. 2026-07-16 SOURCE-FRESHNESS CORRECTION: the '22<=a(17)<=23 open' framing was stale — f(17)=22 since Parsons 1975 (confirmed by Boza 2026 table and by counting against ex(22;C4)=52). The 1,000,000-conflict UNKNOWN on n=17,m=22 was a search for a witness that provably cannot exist. Frontier tracking must consult the survey literature (Radziszowski DS1, Boza) not only the OEIS b-file. 2026-07-17 RETRACTION: the f(39)=46 equality claim was caught by our own novelty gate BEFORE external submission — lower bound 46 was WSR 2015; Boza's upper '/46' is a convention error; DS1.18 says open in {46,47}. Witness stands as a re-derivation certificate. LESSON: verify against DS1 (the maintained survey), and never trust a single survey table for an equality claim.","evidence":{"status":"verified","checked_at":"2026-07-13T07:30:00Z","digest":"sha256:6fad85db5cc5925f5a5894446c56720433065652fecb284a1340262a70a914d3","artifact_path":"certificates/erdos-552/witnesses.json","artifact_sha256":"6fad85db5cc5925f5a5894446c56720433065652fecb284a1340262a70a914d3","verifier_path":"certificates/erdos-552/verify.py","verifier_sha256":"b31ab2d4950825fbb7270f986ada54652d152885fb158eaa3209178bfaa4bcb2","claims":["R(C4,K1,12)=17","R(C4,K1,13)=18","R(C4,K1,14)=19","R(C4,K1,15)=20","R(C4,K1,16)=21","R(C4,K1,17)>=22"],"independent_review":"CHRONOS hostile-referee replay passed n=12; a separate direct C4 enumerator replayed all six graphs; the bundled verifier checks every graph identically."},"compute":{"schema":"p42-atlas-compute-v1","status":"completed","method":"edge SAT with exact C4 clauses and sequential-counter minimum-degree constraints","solver":"PySAT 1.9.dev5 / CaDiCaL 1.9.5","hardware":"local Apple workstation; single solver process","parameter_region":{"n_min":12,"n_max":17,"m_min":16,"m_max":22},"coverage":[{"axis":"n","start":12,"end":16,"status":"CERTIFIED","result":"Exact values 17,18,19,20,21","artifact_sha256":"6fad85db5cc5925f5a5894446c56720433065652fecb284a1340262a70a914d3"},{"axis":"m","start":21,"end":21,"status":"CERTIFIED","result":"21-vertex witness proves R(C4,K1,17) >= 22","artifact_sha256":"6fad85db5cc5925f5a5894446c56720433065652fecb284a1340262a70a914d3","where":{"n":17}},{"axis":"m","start":22,"end":22,"status":"EXCLUDED","result":"Cell closed by literature, not by this compute: min-degree-5 on 22 vertices needs 55 edges > ex(22;C4)=52 [OEIS A006855], so no witness exists; a(17)=22 (Parsons 1975)","where":{"n":17}}],"result":"SAT witnesses closed n=12..16 and improved n=17 to 22..23; n=17,m=22 remained UNKNOWN","limits":"n=17 endpoint used a 1,000,000-conflict budget; UNKNOWN is not an exclusion."},"verdict":"PARTIAL. The asymptotic prize headline remains analytic and is not claimed. Exact certificates close n=12..16; n=17 remains open in 22..23.","lane":"SAT+DRAT-nonexistence","board_class":"READY","board_class_reason":"witness = C4-free red graph with minimum degree >= m-n; codegree and degree checks are exact and cheap; the next open cell a(17) is witness-improvable from the certified m=21 seed","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/552","links":{"oeis":["A006672"],"arxiv":[]},"frontier":{"summary":"Repository-certified: a(12..16) = 17,18,19,20,21. a(17) = 22 CLOSED (Parsons 1975: R(C4,K1,q^2+1)=q^2+q+2 at q=4; independently, the certified 21-vertex witness gives a(17)>=22 and min-degree-5 on 22 vertices needs 55 edges > ex(22;C4)=52 [OEIS A006855], so a(17)<=22). CORRECTION 2026-07-17: the earlier 'a(39)=46 NEW VALUE' claim is RETRACTED — DS1 rev.18 (Apr 2026) Table IVa lists a(39)='46-47' OPEN (lower 46 = Wu-Sun-Radziszowski 2015 Cons.5; upper 47 = Dybizbanski-Dzido); Boza arXiv:2409.12770's '/46' cell is an off-by-one wheel-convention error vs its own source. Our 45-vertex witness (certificates/erdos-552-f39/) re-derives the 2015 lower bound. Open deciders: a(39) via SAT cell (46-vtx min-deg-7: UNSAT=>46, SAT=>47); a(42) via (49,7); a(44) via (51,7). Consult DS1, not single-survey tables."},"p42_slug":"c4-star-ramsey-a17"},{"id":564,"title":"Doubly-exp lower bound for 2-color 3-uniform Ramsey R_3(n)","prize":null,"statement":"Let R_3(n) be the least m so that every 2-coloring of the triples of an m-set contains a monochromatic complete 3-uniform K_n^(3). Erdos-Hajnal-Rado give 2^{cn^2} < R_3(n) < 2^{2^n}. Question ($500): is there c>0 with R_3(n) >= 2^{2^{cn}} (i.e. is the growth genuinely doubly-exponential)?","finite_object":"The asymptotic question has NO finite witness. The in-class finite sub-problems are exact/record small values R(n1,n2;3): a 2-coloring of the triples of an m-set with no red K_{n1}^(3) and no blue K_{n2}^(3) (lower-bound witness), or a DRAT proof that none exists (upper bound).","current_record":"Headline (2-color diagonal, asymptotic): OPEN; gap 2^{cn^2} < R_3(n) < 2^{2^n} (Erdos-Hajnal-Rado 1965). Doubly-exp lower bound is proven only for >=4 colors (Erdos-Hajnal-Mate-Rado); the 2-color case is the missing piece and is called one of the most intriguing open problems in the area (Conlon-Fox-Sudakov, Mubayi survey). Finite sub-problem records: R(4,4;3)=13 EXACT (lower Isbell 1969, equality McKay-Radziszowski 1991 -- the first-ever exact hypergraph Ramsey value). Off-diagonal R(4,5;3): >=24 (Isbell 1983) -> >=33 (Exoo 1998) -> >=35 via SAT-constructed coloring on 34 vertices (Contributions to Discrete Math, ~2018); upper bound believed far larger. R(5,5;3) small values essentially unknown; large-n lower bounds come from algebraic 'pair/stepping-up' constructions, not search.","beatable":"WALL","beatable_reason":"Two-layer wall. (1) The $500 headline is asymptotic and provably NOT finite-resolvable: closing 2^{cn^2} vs 2^{2^n} requires a 2-color stepping-up lemma, which is exactly the missing THEOREM (Erdos-Hajnal stepping-up only fires for uniformity>=3 AND colors>=4). No finite computation touches it. (2) The one genuinely search-shaped finite frontier -- the lower bound for R(4,5;3), record >=35 from a good 2-coloring on 34 vertices -- was itself set by a SAT solver with class/symmetry decomposition, the identical tooling we'd bring. This is the R(5,5)-witness lesson verbatim: the record IS the wall for our tool. Improving to >=36 needs a good coloring on 35 vertices (C(35,3)=6545 vars, ~377k clauses of a notoriously hard Ramsey SAT family); the 2018 authors already needed a structured class-decomposition ansatz just to reach 34, and raw symmetric SAT does not beat them. R(4,4;3)=13 is already closed. Our edge over the published SAT work is essentially nil beyond a bigger farm + newer solver + better symmetry breaking, none of which is known to crack the next vertex. A win here would also be non-prize (a minor off-diagonal record), not a resolution of #564.","fit_score":7,"impact_score":4,"impact_reason":"The named problem is a prestigious $500 Erdos problem and a flagship open question in hypergraph Ramsey theory -- but that headline is unreachable by us. The only slice our tools could touch is nudging an off-diagonal small-case lower bound (e.g. R(4,5;3) 35->36), which is a minor, non-prize, incremental record with no bearing on the asymptotic conjecture and zero real-world relevance. High prestige of the problem, low significance of our reachable deliverable.","verifier":"Poly-time exact: given a coloring of the C(m,3) triples, scan all C(m,4) 4-subsets (no red K_4 = not all 4 triples red) and all C(m,5) 5-subsets (no blue K_5). O(m^5) checks, trivial. Nonexistence certifiable via SAT UNSAT + DRAT, exactly as R(5,5).","attack":"If attempted anyway (best shot, low expected yield): target R(4,5;3) >= 36 by SAT-searching for a good 2-coloring on 35 vertices. Encoding: C(35,3)=6545 booleans (one per triple); for each of C(35,4)=52360 4-subsets a clause forbidding all-red; for each of C(35,5)=324632 5-subsets a clause forbidding all-blue (~377k clauses). Break symmetry with a structured ansatz rather than naive lex: seed a vertex-transitive/affine construction (cyclic Z_35 or GF-based, as Exoo/CDM did) so the free search is over class-colorings, then Kissat/CaDiCaL portfolio across ~100 cores with many random seeds and cube-and-conquer splitting. Similarly probe R(5,5;3) from a cyclic seed. Rough compute: days-to-weeks of a 100-core farm per target with no guarantee of SAT; if UNSAT on the structured subspace it proves nothing global. DRAT only meaningful if aiming to certify a tight upper bound, which is out of reach here (upper bounds are far above the constructions).","our_edge":"None decisive. Generic: ~100-core farm + modern SAT (Kissat/CaDiCaL) + DRAT pipeline reused from R(5,5). But the standing record was set with the same class of tools, and the binding constraint is a structured CONSTRUCTION (algebraic pair/stepping-up ansatz), which is a math contribution, not a compute contribution -- outside our verifier-first farm's comparative advantage.","verdict":"PARTIAL -- headline is asymptotic and needs a new 2-color stepping-up theorem (NONE for the $500 question); the in-class finite sub-problem (exact/record small values R(n1,n2;3)) maps cleanly onto our SAT-witness + poly-verifier + DRAT farm, but the diagonal base case R(4,4;3)=13 is already closed and the only live search frontier (R(4,5;3)>=35, R(5,5;3)) had its record set by the same SAT tooling and sits at the wall. Machinery fit is high; beatability is low. Recommend NO attack: high effort, low-value non-prize record, record set by identical tools. Log as a clean PARTIAL/WALL, exactly parallel to R(5,5) (upper-bound NONE, witness at the wall).","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = 2-coloring of triples on 35+ vertices; O(m^5) scan of 4-/5-subsets, exact, cheap; R_3(4,5;3) >= 36 witness-improvable","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/564","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"R_3(4,5;3) >= 35 (SAT class-decomposition record)"}},{"id":582,"title":"Folkman number Fe(3,3;4)","prize":"$100 (Erdős, for the original existence — already PROVED-in-Lean); separately Graham's $100 for proving/disproving N ≤ 100.","statement":"Does there exist a K4-free graph G such that every 2-coloring of its edges yields a monochromatic K3? Existence is settled (Folkman 1970); the live frontier is the smallest such G — the edge Folkman number N = Fe(3,3;4).","finite_object":"A finite graph G on n vertices that (a) contains no K4 and (b) arrows (K3,K3) edge-wise. Beating the record = exhibiting such G with n < 786; the lower-bound direction = certifying no such G exists on n ≤ 21, 22, ...","current_record":"21 ≤ Fe(3,3;4) ≤ 786. Upper bound 786 = Lange, Radziszowski & Xu 2012 (explicit algebraic/Ramsey-based construction; prior chain: Lu 9697 in 2007, Dudek-Rödl 941 in 2008). Lower bound 21 = Bikov & Nenov 2020 (heavy independence-number computation on (3,3)-Ramsey graphs). Graham offers $100 for N ≤ 100; Radziszowski-Xu speculate N ≤ 127. No improvement to 786 in 14 years as of July 2026.","beatable":"UNKNOWN","beatable_reason":"The verifier is a perfect R(5,5)-style fit (K4-free + DRAT-certified UNSAT arrowing), but the frontier that matters — the 786 upper bound — is a CONSTRUCTION problem, not a blind-search problem, and it has resisted 14 years of expert effort. Two paths: (A) UPPER BOUND / find smaller G — blind local-search/SA is a WALL: fitness is a coNP arrowing query (each evaluation is a full UNSAT solve), there is no cheap \"distance-to-arrowing\" gradient, and the K4-free graph space at hundreds of vertices is hyper-astronomical; every known record came from algebraic structure, not search. (B) LOWER BOUND / push 21→22 — a WALL: requires enumerating ALL K4-free graphs on 21 vertices and exhibiting a good coloring for each; Bikov-Nenov's 20→21 already needed bespoke independence-number pruning, and the enumeration blows up past feasibility. The ONE live crack: the June-2025 \"Hermitian unital\" H_q construction (Mulrenin, arXiv 2506.14942) proposes structured small candidates (H_3 = 63 vertices, H_4 = 208 vertices) that may contain a K4-free arrowing subgraph far below 786 — that converts the problem from \"invent a construction\" into \"verify a proposed structured candidate,\" which IS our machinery. But the paper is exploratory and no verified sub-786 graph has been extracted. So the honest call is UNKNOWN: a genuine but unproven conditional crack riding an external construction, not a movable-by-generic-compute record.","fit_score":6,"impact_score":6,"impact_reason":"Prizes are modest ($100 Erdős, $100 Graham), but field significance is high: Folkman numbers are a famous named problem with a dedicated \"most wanted Folkman graph\" literature; beating the 14-year-old 786 bound, or hitting Graham's ≤100 target, would be a celebrated combinatorics result. Real-world relevance is negligible. The Lean-verified existence means the headline ($100 Erdős) is already claimed — only the quantitative record remains.","verifier":"K4-free: trivial poly check (enumerate 4-subsets or clique-free test). Arrowing: encode \"there EXISTS a 2-edge-coloring with no monochromatic triangle\" as SAT (one Boolean per edge; for each triangle two clauses forbidding all-red and all-blue). G arrows iff that formula is UNSAT. UNSAT is DRAT-certifiable. Cost: one large CDCL UNSAT per candidate — comparable scale to the R(5,5) window-UNSATs we already ran; hours-to-days for a ~200-800-vertex graph, DRAT check linear in proof size.","attack":"(1) Build the arrowing verifier: SAT-encode \"no good 2-coloring\", solve with a modern CDCL solver (CaDiCaL/Kissat), emit DRAT — reuse of the exact R(5,5) pipeline. (2) Generator is the bottleneck; do NOT blind-SA unstructured graphs. Instead take the 2025 Hermitian-unital seeds (H_3 = 63v, H_4 = 208v and their triangle systems) as structured candidates, then run SAT/MaxSAT-guided vertex/edge deletion that greedily shrinks the seed while re-verifying K4-free + arrowing (UNSAT) at each step — a verifier-in-the-loop minimization rather than a global search. (3) In parallel, run the known 786 LRX graph through the same minimizer to test whether local surgery + DRAT re-cert shaves vertices. Compute: each arrowing check on a 60-800-vertex candidate is one heavy DRAT-UNSAT (hours-days); a minimization sweep over the seed family = weeks on a ~100-core farm — feasible ONLY because the generator is algebraic, not blind. Pushing the lower bound (enumerate all 22-vertex K4-free graphs) is infeasible on this budget — skip.","our_edge":"We have the exact verifier already built and battle-tested from R(5,5): SAT-encode arrowing, DRAT-certify UNSAT, run at scale on the CPU farm. That is a real, specific edge for VERIFYING a candidate and producing a certificate a referee can check. What we lack is a generator — our farm is verifier-first with blind search, and this record has no blind-search on-ramp. Edge beyond generic compute exists ONLY if we couple the verifier to the external 2025 geometric construction as the seed source.","verdict":"STRONG-in-structure / WALL-in-practice. Structurally this is textbook STRONG: a finite object (a specific graph) beating a concrete record (786), with a poly-time + DRAT-certifiable exact verifier identical to our R(5,5) tooling, and the frontier is a NUMBER. But adversarially, the record is expert-hardened against exactly the tools we bring: the upper bound needs an algebraic construction (blind SA/local-search dies on coNP fitness) and the lower bound needs infeasible graph enumeration. It is NOT a generic-compute-movable record. The only honest path to a win is a verifier-in-the-loop minimization seeded by the June-2025 Hermitian-unital candidates (H_3=63v / H_4=208v) — high-value if it lands (smashes 786, possibly claims Graham's $100) but speculative and dependent on someone else's construction. Recommend: LOW-PRIORITY WATCH, not an active farm target; revisit if the Mulrenin seeds yield an extractable sub-786 candidate to certify.","lane":"witness-local-search","board_class":"HEAVY","board_class_reason":"witness (K4-free) is cheap, but adjudicating 'G arrows (3,3)' costs one large CDCL UNSAT + DRAT check per candidate - optimistic-oracle tier (R1 fails)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/582","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"21 <= Fe(3,3;4) <= 786 (Lange-Radziszowski-Xu 2012); Graham's $100 for a K4-free graph <= 100 vertices arrowing (3,3)"}},{"id":588,"title":"Erdős #588 — f_k(n)=o(n²) for k≥4 (orchard-type k-rich lines)","prize":"$100 (Erdős, [Er84])","statement":"Let f_k(n) be minimal such that any n points in R² with no k+1 collinear span at most f_k(n) lines containing ≥k points. Is f_k(n)=o(n²) for k≥4? ($100, open.) Finite in-class sub-problem: exact small-n orchard-type extremal values — max lines of exactly k points with no k+1 collinear (OEIS A006065 for k=4, A008997 for k=5).","finite_object":"A point configuration in R² (equivalently an incidence/line-arrangement structure) of n points, no k+1 collinear, maximizing the count of lines with exactly k points — the orchard-type extremal config for a given (n,k).","current_record":"Headline conjecture OPEN (no proof either way); best construction n^{2-c/√log n} collinear k-tuples, Solymosi–Stojaković 2013 (still o(n²), so consistent with conjecture). Finite tables: A006065 (k=4) proven optimal up to n=20, value 23 — Zhao Hui Du, distributed exhaustive incidence enumeration + Gröbner realizability, ~2010; lower-bound constructions to n=25 (Friedman; Ed Pegg n=22, 2020). A008997 (k=5) sparse, Friedman bounds to n=21.","beatable":"WALL","beatable_reason":"Continuous-geometry extremal problem, not a SAT/ILP/MIS instance. Upper-bound frontier (n≥21, k=4) is bottlenecked by abstract line-incidence enumeration + Gröbner-basis realizability — the exact specialized comp-alg-geom pipeline the record-holder already built and optimized, explicitly prohibitive beyond n≈20; our farm doesn't do this and has no edge. Lower-bound/witness direction is killed by measure-zero collinearity: float SA/local-search cannot land on exact k-fold incidences, and record configs require algebraic (cubic-curve/group-law) coordinates. Grid restriction (A172992) is SAT/ILP-able but yields strictly weaker-than-record configs. Same posture as R(5,5)'s upper-bound side: computational-looking but at the feasibility wall for the tools we own.","fit_score":3,"impact_score":4,"impact_reason":"Classic, charming Erdős problem but only $100; asymptotic headline is a niche incidence-geometry question with essentially no real-world relevance. Extending an orchard table by one term (n=21, k=4) would be a real but small, non-prize-claiming contribution to the sibling problem #669, not to #588's headline.","verifier":"Given n points with exact rational/algebraic coordinates: check every pair-generated line hits ≤k points (no k+1 collinear) and count lines with ≥k points. O(n²log n)–O(n³), trivially cheap. Poly-time exact — fits verifier-first, but only verifies; it does not constrain the continuous search.","attack":"Two candidate lines, both walled. (a) Upper bounds n≥21,k=4: exhaustively enumerate combinatorial line-arrangements, prune by projective symmetry (nauty), test real/complex realizability via Gröbner bases (Maxima/PARI). This replicates the 2010 record tool; compute grows super-exponentially, no advantage from a 100-core SAT/MIS farm. (b) Lower-bound witness: optimize over R^{2n} for more k-lines than Friedman's best (n=21–25 k=4; n=16–21 k=5). Requires exact algebraic coordinates — SA/MIS/ILP over floats or a bounded grid give weaker configs than existing algebraic constructions. No realistic compute path to a new record with our toolkit.","our_edge":"None beyond generic compute. Our SAT/DRAT/MIS/SA/LP farm does not map to Gröbner-realizability enumeration (the record pipeline) nor to exact-algebraic construction (the record constructions). We hold no comp-alg-geom instrument that beats Zhao Hui Du's 2010 stack.","verdict":"PARTIAL headline (asymptotic o(n²), inherently non-finite — Solymosi–Stojaković construction stays o(n²) so the conjecture is still open, not computationally decidable), WITH a genuine finite in-class sub-problem: the orchard-type exact tables A006065/A008997. But that sub-problem is a WALL for our machinery — the verifier is cheap and clean, yet the search is continuous-coordinate comp-alg-geom (Gröbner realizability for upper bounds; exact algebraic constructions for witnesses), not SAT/ILP/MIS/SA. Measure-zero collinearity defeats float search; grid restriction is strictly weaker than standing records; the record pipeline is precisely the specialized tool we lack. No edge. Do not attack.","lane":"wall","board_class":"NONE","board_class_reason":"same ExistsR/collinearity obstruction as #101 (R3 fails); the continuous search is unconstrained by the verifier","wall_reason":"same ExistsR/collinearity obstruction as #101 (R3 fails); the continuous search is unconstrained by the verifier","erdos_url":"https://www.erdosproblems.com/588","links":{"oeis":["A006065","A008997"],"arxiv":[]}},{"id":687,"title":"Erdős #687 — estimates for Y(x), the primorial covering optimum","prize":"$1000 (for the asymptotic estimate; unclaimable by finite computation)","statement":"Y(x) = max y such that one can choose a residue class a_p mod p for every prime p≤x with every integer in [1,y] hitting at least one a_p. Give good estimates for Y(x); in particular prove Y(x)=o(x²) or Y(x)≪x^{1+o(1)}. This is exactly the Jacobsthal function of the primorial P(x)=∏_{p≤x}p (equivalently, the longest run of consecutive integers each divisible by some prime ≤x). Headline asks for an ASYMPTOTIC bound. Known: Y(x)≪x² (Iwaniec 1971, upper) and Y(x)≫x·logx·logloglogx/loglogx (Ford–Green–Konyagin–Maynard–Tao 2018, lower).","finite_object":"A covering witness: an assignment {a_p mod p : p≤x} together with an interval [u, u+V−1] all of whose integers are congruent to some a_p. Equivalently a run of V consecutive integers each divisible by a prime ≤x. This certifies Y(x) ≥ V. Verified against the OEIS table A048670 = g(primorial) and its interval-endpoint companion A058989.","current_record":"a(1..64) of A048670 exact; latest a(62)–a(64) by Andrzej Bozek, 2021, Google-Cloud-supported, via Ziller–Morack pruned exhaustive search (Costello–Watts upper bounds). No extension 2023–2026.","beatable":"WALL","beatable_reason":"The record is held by precisely our own method class — massively parallel exhaustive search with domain-specific pruning — run at Google-Cloud scale (Bozek 2021). The next term a(65) sits at/beyond the feasibility frontier: the search over residue assignments mod the primorial grows super-exponentially, and the last three terms already required cloud-scale compute exceeding a ~100-core farm. Generic SAT/ILP/MIS encodings have NO edge over the hand-tuned Ziller–Morack pruning (they'd be dramatically worse), and there's no clean DRAT-certified nonexistence hook because the hard direction is the completeness of a bespoke search, not a CNF UNSAT. This is the R(5,5)-witness situation: computationally-shaped but already pinned to the wall by the same tools we'd bring. Lower-bound witnesses are trivially verifiable but can't \"beat\" an already-exact record without also doing the (infeasible) matching upper bound.","fit_score":3,"impact_score":4,"impact_reason":"$1000 prize but it attaches to the asymptotic estimate (o(x²) / x^{1+o(1)}), which finite computation cannot claim — a table extension wins no prize. Field significance is real but moderate: the object is the Jacobsthal function of primorials, tied to large prime gaps (Ford–Green–Konyagin–Maynard–Tao) and Erdős covering systems; a new term would be a citable but minor OEIS/Math.Comp.-level contribution. Real-world relevance low.","verifier":"Lower-bound side is a POLY-TIME exact verifier: given a claimed run [u,u+V−1] and the residues, check each of the V integers is divisible by some prime ≤x — O(V·π(x)). Cheap. BUT the exact value / OEIS-table entry requires the UPPER-bound side: certify NO run of length V+1 exists over all ∏_{p≤x}p residue choices — an exhaustive/pruned nonexistence search whose cost is the whole difficulty (no compact DRAT-style certificate is standard here; correctness rests on the specialized search's completeness).","attack":"If attempted (not recommended): to push a(65)=Y(p_65=313), run the Ziller–Morack framework — fix a per-prime residue search with the strong \"reach/interval-cover\" pruning and Costello–Watts upper-bound cutoffs, distributed over the farm. Realistic compute: a(64) needed Google-Cloud scale (plausibly 10^4–10^5 core-hours with the specialized code); a(65) is meaningfully harder. A ~100-core farm running a from-scratch SAT/ILP encoding would not reach even a(60) in feasible time. No credible path to a novel record with our generic verifier-first stack.","our_edge":"None beyond generic compute — and our compute is smaller than the incumbent's. The one clean map to our machinery (poly-time lower-bound witness checking) is the EASY half; the record lives entirely in the hard nonexistence half where the specialized incumbent code dominates a generic SAT/ILP approach.","verdict":"PARTIAL — The headline is asymptotic and NONE for finite computation (a new theorem is the only path to the prize; the two-sided bounds are analytic). There IS a genuine in-class finite sub-problem: Y(x) is for each x an exactly-computable covering-system optimum, and its table A048670 is extendable — a real (non-prize) contribution. But adversarially the frontier is a WALL: the current record (a(64), Bozek 2021) was set by cloud-scale runs of the same exhaustive-search-with-pruning method we would use, the next term is at the feasibility edge, our farm is smaller, and generic SAT/ILP gives us no edge over the specialized Ziller–Morack algorithm. Verdict: PARTIAL in-class, but not attackable by us — do not pursue.","lane":"exact-backtracking","board_class":"HEAVY","board_class_reason":"next A048670 term requires the upper-bound side: certified nonexistence over all residue choices - pruned-exhaustion receipt, not a witness","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/687","links":{"oeis":["A048670"],"arxiv":[]},"frontier":{"summary":"A048670 exact to a(64) (Bozek 2021, Google-Cloud pruned exhaustion); next term = pruned-exhaustion certificate claim"}},{"id":703,"title":"Forbidden-intersection family T(n,r) (Erdős #703)","prize":"$250 (already awarded/solved — problem marked PROVED)","statement":"T(n,r) = max size of a family F⊆2^[n] with |A∩B|≠r for all A,B∈F. Erdős asked to estimate T(n,r) for r≥2, in particular whether T(n,r)<(2-δ)^n for εn317 on the CPU farm, hoping to FIND an extension and thereby refute it as a candidate. Compute is trivial (O(q²) per q, minutes for thousands of q). But this is (a) already pushed past q=317 and withdrawn, (b) expected to fail since experts believe {0,1,3,11} is a true counterexample, and (c) non-terminating as a proof — a negative sweep proves nothing and a positive hit is a minor note, not a theorem or prize.","our_edge":"None beyond generic compute — and even that is spent: the finite search range is already exhausted (q≤317) and superseded by Mueller's stronger algebraic result. The clean per-q verifier is genuinely in our wheelhouse, but the question it feeds is unbounded, so the verifier cannot close it.","verdict":"PARTIAL — genuine finite sub-problem in-class (per-q Singer embedding of {0,1,3,11}, exact poly-time verifier), but the headline is already SOLVED (prize claimed, Lean-verified) and the residual size-4 question is intrinsically infinite over all prime powers, so finite computation cannot settle it. The one finite lever (extend the embedding sweep past q=317) is already exhausted, superseded by Mueller's algebraic proof, and non-prize-claiming. Effectively a WALL for our witness-or-nothing farm; recommend PASS.","lane":"wall","board_class":"NONE","board_class_reason":"headline DISPROVED, Lean-verified (Alexeev-Mixon, PNAS 2025); nothing movable remains","wall_reason":"headline DISPROVED, Lean-verified (Alexeev-Mixon, PNAS 2025); nothing movable remains","erdos_url":"https://www.erdosproblems.com/707","links":{"oeis":[],"arxiv":["2510.19804","2604.25214"]}},{"id":708,"title":"g(n) divisibility over windows of consecutive integers","prize":"$100 (Erdős), lowest tier","statement":"g(n) = minimal g such that for every A ⊆ [2,∞)∩ℕ with |A|=n and every window I of max(A) consecutive integers, there is B ⊆ I with |B|=g and ∏_{a∈A} a | ∏_{b∈B} b. Conjecture: g(n) ≤ (2+o(1))n, or perhaps even g(n) ≤ 2n.","finite_object":"A counterexample witness (A, I) forcing minimal valid B > 2n; finite and exhaustively checkable, but almost certainly nonexistent at reachable scale.","current_record":"No tracked numeric record and no small-value table (OEIS field empty). Erdős–Surányi (1959, [ErSu59]) posed it and established g(n) well-defined/finite; re-stated in Erdős 1992 ([Er92c],[Er92e]). The conjectured bound g(n) ≤ (2+o(1))n (or ≤2n) is OPEN. erdosproblems.com marks it explicitly \"cannot be resolved with a finite computation.\" No 2023–2026 movement found in searches. (Exact best-known upper bound from ErSu59 not confirmable via available sources — but classification does not hinge on it.)","beatable":"WALL","beatable_reason":"The prize question is asymptotic (∃-o(1)) and needs a theorem — not finitely resolvable. Exact g(n) values are NOT computable either: the upper-bound direction ranges over infinitely many A with unbounded max(A), and no proven cap reduces this to a finite check (proving the cap IS the theorem). The only finitely-verifiable handle is a counterexample to the strong ≤2n form, but (a) there is no tracked record to advance, (b) Erdős floats 2n as plausibly true so the search almost certainly returns nothing, (c) a ≤2n counterexample would not even touch the asymptotic prize form, and (d) forcing minimal-B>2n needs m>2n where the 2^m exhaustive verifier explodes.","fit_score":3,"impact_score":2,"impact_reason":"$100 prize (lowest Erdős tier), obscure 1959 number-theory problem, empty OEIS, no recent literature activity, no real-world relevance. Even a successful finite result (a ≤2n counterexample) would be a curiosity that leaves the asymptotic prize question untouched.","verifier":"For a candidate lower-bound witness (A, I): exact check that no B ⊆ I of size ≤ k has ∏A | ∏B — exhaustive over 2^m subsets (m = max(A)) with trivial integer-divisibility tests. Cheap for small m, exponential as m grows. NOTE: the UPPER bound g(n)≤k has NO finite verifier — it quantifies over the infinite family of n-subsets A with unbounded max(A), with no proven a priori cap.","attack":"Only conceivable finite attack: enumerate A ⊆ [2,m] with |A|=n for small n and m up to a few dozen; for each, over all windows I of length m, ILP/exhaustively compute the minimal B ⊆ I meeting the prime-valuation cover ∏A|∏B (a covering/set-cover over prime powers), and flag any instance with minimal |B| > 2n. Rough compute: trivial CPU-minutes at small n/m, but the reachable range (m ≲ 30) is far below where any counterexample could plausibly first appear, and Erdős's own phrasing suggests none exists — so expected yield is zero. Not worth farm time.","our_edge":"none beyond generic compute","verdict":"NONE (weak-PARTIAL at most). Headline is asymptotic → needs a theorem. Exact small g(n) values are NOT computable (unbounded-A upper quantifier, no proven cap), contra the triage note. The sole finite handle — an exhaustive counterexample-to-2n search — mechanically fits our farm but has no tracked record to push and near-certain null yield, and would not resolve the prize form even on success. Recommend PASS.","lane":"wall","board_class":"NONE","board_class_reason":"no tracked record, and the upper-bound direction quantifies over unbounded sets with no finite verifier","wall_reason":"no tracked record, and the upper-bound direction quantifies over unbounded sets with no finite verifier","erdos_url":"https://www.erdosproblems.com/708","links":{"oeis":[],"arxiv":[]}},{"id":712,"title":"Hypergraph Turán density of K_k^r (tetrahedron 5/9)","prize":"$500","statement":"Determine, for k>r>2, the Turán density lim ex_r(n,K_k^r)/C(n,r): the largest fraction of r-edges on n vertices with no k-set carrying all C(k,r) possible r-edges. Flagship case r=3,k=4 (the tetrahedron): is π(K_4^3)=5/9?","finite_object":"For a fixed n: a maximum K_k^r-free r-graph on n vertices (an exact small-case value ex_r(n,K_k^r)), e.g. a max tetrahedron-free 3-graph on n vertices. The headline density is NOT a finite object.","current_record":"Tetrahedron K_4^3: LOWER 5/9≈0.55556 (Turán 1941 construction; exponentially-many non-isomorphic extremal families later found by Brown, Kostochka, Fon-der-Flaass, Frohmader). UPPER π(K_4^3)≤0.561666 by Razborov (flag algebras / SDP, 2010), sharpened to ~0.5615 by Baber–Talbot (flag algebras, ~2011). Gap ~0.006 still open; no closure 2023–2026 (recent work is on ℓ2-norm variants and uniform Turán density, not this). Exact ex(n,K_4^3) known only for small n; NOT in OEIS.","beatable":"WALL","beatable_reason":"The $500 frontier is the ASYMPTOTIC density — the page itself states it cannot be resolved by finite computation, and it is one of the oldest open problems in extremal combinatorics (80+ yr). Its three levers are all walls for us: (1) UPPER bound is a flag-algebra SDP — same certificate class as our PNT/autocorrelation work — but it is already run to feasibility limits (order-6/7 flags) and there is a known flag-algebra barrier suggesting the method alone cannot reach 5/9; shaving the 4th decimal resolves nothing. (2) LOWER bound is capped at the conjectured-optimal 5/9, with matching constructions already known — no config can beat it. (3) The only genuinely finite, verifier-first slice — exact ex(n,K_4^3) for a new n via SAT/ILP — is a footnote with no tracked record and no OEIS entry, and even that hits the wall fast: near-optimum solution density is high and the automorphism group is huge, so the UNSAT/optimality proofs (not the SAT witnesses) are the bottleneck, and the outcome never touches the density. Nothing here converts our farm into a defensible record.","fit_score":4,"impact_score":7,"impact_reason":"The tetrahedron conjecture is a top-tier, ~85-year-old open problem in extremal combinatorics with an Erdős $500 prize and enormous field prestige — but real-world relevance is low, the prize is modest, and the impact of the ONLY slice we could touch (a footnote exact value) is essentially zero. High headline significance, near-zero reachable significance.","verifier":"Poly-time exact: given an r-graph, test all C(n,k) k-subsets are non-complete → O(n^k) (O(n^4) for the tetrahedron). For an exact-value optimality claim, a SAT decision \"∃ K_k^r-free r-graph with ≥m edges\" yields a DRAT nonexistence certificate at m+1. Cheap either way.","attack":"If one insisted on the finite slice: decision SAT \"∃ K_4^3-free 3-graph on n vertices with ≥m edges\" — C(n,3) boolean edge vars, C(n,4) clauses forbidding any 4-set being complete, a ≥m cardinality constraint; binary-search m; UNSAT at m+1 → exact ex(n,K_4^3) + DRAT cert. Instances are small (n=16 → 560 vars, 1820 base clauses) so witnesses are easy; the cost is the boundary UNSAT proof under a large symmetry group (needs SBP/orbit-pruning), ~100-core-days for a single new n — for a value nobody tracks. Not worth farm time.","our_edge":"None beyond generic compute. Our SDP-certificate machinery is the right shape for the upper bound but the SDP is already at the community's feasibility+theoretical wall; our SAT/ILP+DRAT machinery is the right shape for exact small values but there is no record to take.","verdict":"PARTIAL-but-effectively-NONE. Headline = pure asymptotic density, needs a new theorem, explicitly non-finite; upper bound is a flag-algebra SDP already at the wall, lower bound is capped at the conjectured-optimal 5/9. There IS a genuine in-class finite sub-problem — exact small ex_r(n,K_k^r) via SAT/ILP with an O(n^k) verifier, mechanically a perfect fit for our farm — but it is an untracked, non-prize footnote that (a) never moves the density and (b) hits an optimality-proof/symmetry wall quickly. Verifier-first mechanics fit; the target does not. SKIP.","lane":"LP/SDP-certificate","board_class":"HEAVY","board_class_reason":"movable claim = an improved flag-algebra SDP dual certificate - exact-rational certificate check, heavy and specialist (LP/SDP lane)","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/712","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"5/9 <= pi(K_4^3) <= 0.561666 (Razborov flag-algebra SDP 2010); movable claim = a better SDP dual certificate"}},{"id":720,"title":"Size Ramsey numbers of paths & cycles","prize":"$100 (already awarded — problem solved by Beck)","statement":"Erdős–Faudree–Rousseau–Schelp asked whether the size Ramsey number satisfies R̂(P_n)/n→∞ and R̂(P_n)/n²→0, and R̂(C_n)=o(n²). Here R̂(G)=min edges m of a host graph H such that every 2-colouring of E(H) contains a monochromatic G.","finite_object":"A host graph H (with m edges) that \"arrows\" the target (H → G): every red/blue edge-colouring yields a monochromatic path/cycle. The exact value R̂ is the minimum such m over all host graphs.","current_record":"Headline: PROVED — Beck 1983 [Be83b], R̂(P_n)≪n and R̂(C_n)≪n (linear, stronger than asked); constant sharpened to <137n (Dubroff–et al. / Ryerson line). $100 already claimed. Exact small values (nauty + backtracking): r̂(P₄,P₅)=10, r̂(P₅,P₅)=11, r̂(P₄,P₆)=14, r̂(C₄,C₅)=19, r̂(C₆,C₆)=26 — Li, Fahad, Zhou, Yang, Frontiers in Physics 8:350 (2020). SAT/SMS (Kissat) used for related small Ramsey values 2022–2026, but not to extend diagonal R̂(P_n)/R̂(C_n) beyond tiny n.","beatable":"WALL","beatable_reason":"The headline is PROVED and thus not prize-movable: Beck (1983) proved the far stronger linear bounds R̂(P_n)≪n and R̂(C_n)≪n (constant later improved to <137n), fully settling both the →∞ and o(n²) questions. The only live finite sub-problem is pinning EXACT small size-Ramsey values, and that frontier is already set by exactly our tools (nauty graph-generation + backtracking; recent Kissat/SMS SAT). Determining R̂ exactly is Σ₂/Π₂-hard: an OUTER search over ALL non-isomorphic host graphs H (unbounded, unlike classical Ramsey's fixed K_N — a graph with m≈30–40 edges already has astronomically many candidate hosts) wrapping an INNER coNP-complete arrowing test (Burr 1976). State of the art reaches only ≤6-vertex targets (r̂(C₆,C₆)=26); the next case explodes on both layers. We have no edge over the incumbents beyond generic compute, and the outer host-search — not a fixed board — is precisely why this is a wall, worse than R(5,5). A narrow movable sliver exists (improve a single small UPPER bound by exhibiting one better host H, DRAT-certifiable), but it is non-prize and niche.","fit_score":4,"impact_score":2,"impact_reason":"Headline conjecture is fully resolved (PROVED, $100 claimed), so no prize and no open theorem to close. The residual finite work — exact small values of R̂(P_n)/R̂(C_n) — appears in niche venues (Frontiers in Physics), extends OEIS-style tables at best, and has no real-world relevance. A certified small upper-bound improvement would be a minor note, not a field event.","verifier":"Arrowing check H→G: SAT-encode \"∃ 2-colouring of E(H) with no monochromatic G\", solver returns UNSAT (H arrows) or a witness colouring (H fails). coNP-complete (Burr). UPPER bound = one host H + a single DRAT UNSAT cert: cheap, farm-friendly, our exact pattern. EXACT value additionally needs the LOWER bound: nauty-enumerate every host with ≤m−1 edges and exhibit an avoiding colouring for each — Σ₂-style, super-exponential host count, the binding wall.","attack":"If attempted (not recommended for prize): pick one under-determined small target, e.g. diagonal R̂(P_6) or R̂(C_7) where only an upper bound is published. (1) Local-search / SA over candidate host graphs H of a few dozen edges to find one that arrows the target; (2) certify H→G by encoding avoid-colouring as CNF and running Kissat to UNSAT + emit DRAT. That yields a certified upper-bound improvement. Pinning the EXACT value requires the outer nauty sweep of all smaller hosts (each a SAT call) — feasible only for ≤~6-vertex targets already done; ~100-core farm buys maybe one extra vertex before the host-enumeration blows up. Net: a minor certified upper bound is reachable; an exact new value is not.","our_edge":"None beyond generic compute. The incumbent method (nauty + backtracking, and now Kissat/SMS SAT) IS our machinery; DRAT arrowing certs are a marginal packaging improvement, not a capability edge. No structural asymmetry to exploit as in R(5,5).","verdict":"PARTIAL — headline is asymptotic AND already PROVED (Beck 1983), so NONE for any prize/theorem path; but a genuine finite sub-problem is in-class: exact/near-exact small size-Ramsey values of paths and cycles, with a clean poly-checkable arrowing verifier (SAT UNSAT + DRAT). That sub-problem is a WALL for our tools on the exact-value frontier — the current records were set by the identical nauty/SAT machinery and the next case explodes on both the unbounded host search and the coNP inner check — leaving only a low-impact, non-prize sliver (certify one better host graph = one improved upper bound). Recommend PASS: strictly dominated by open witness-STRONG targets.","lane":"SAT+DRAT-nonexistence","board_class":"HEAVY","board_class_reason":"each exact-value move = host graph + DRAT UNSAT certificate (upper) + enumeration receipt (lower) - certificate-tier adjudication","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/720","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"size-Ramsey exact small values (r-hat(P5,P5)=11, ...); each upper-bound move = one host + one DRAT UNSAT certificate"}},{"id":1029,"title":"R(k)/(k·2^{k/2}) → ∞ (diagonal Ramsey growth)","prize":"$100 (attached to the asymptotic theorem, which is unreachable by finite computation)","statement":"Erdős #1029 ($100): prove R(k)/(k·2^{k/2}) → ∞, where R(k)=R(k,k) is the diagonal Ramsey number. The page itself flags it \"cannot be resolved with a finite computation.\"","finite_object":"In-class finite target = a clique-free Ramsey graph on n vertices witnessing a new diagonal lower bound R(k,k) ≥ n+1 (a 2-coloring of K_n with no monochromatic K_k), i.e. exactly our R(5,5) witness object — or DRAT-style certification of nonexistence at a given n.","current_record":"Headline: asymptotic conjecture, OPEN (only R(k) ≥ 2^{k/2+o(k)}, Erdős 1947 probabilistic; no explicit construction ever reaches exponential). Finite frontier: R(5,5)∈[43,46] — lower 43 = Exoo 1989 (cyclic graph on 42 vtcs, conjectured tight, 656 critical K42-graphs none extending); upper 46 = Angeltveit–McKay 2024. R(6,6)∈[102,160] — lower 102 UNCHANGED since 1966; R(8,8) lower unchanged since 1972. For k≥13 best lowers are deterministic Paley graphs.","beatable":"WALL","beatable_reason":"Every accessible finite record is at the feasibility wall for our exact toolset: (a) R(5,5)≥43 is conjectured tight and we already ran the witness hunt (~$34 farm, no K5-free-both-colors graph on 43 vtcs; cyclic space on prime 43 is small and exhausted, full space astronomically hard); (b) R(6,6)/R(7,7)/R(8,8) lower bounds have resisted 55–60 years of exactly the cyclic/circulant local-search + SAT + simulated-annealing methods we would use, with zero movement; (c) large-k records are Paley-graph deterministic (prime-power fixed, no local-search knob); (d) ALL genuine 2023–2026 progress in this class is asymptotic/analytic (CGMS upper bound; Campos–Jenssen–Michelen–Sahasrabudhe R(3,k) probabilistic lower bound) or lives OFF-diagonal / multicolor (Wesley SAT on R(3,8), book & multicolor Ramsey) — none is a finite diagonal-witness record. Record set by the same tools we'd use = textbook wall, just like R(5,5) itself was.","fit_score":8,"impact_score":6,"impact_reason":"The asymptotic theorem is a famous Erdős target but only $100 and pure analysis — off-limits to us. A finite diagonal record (R(5,5)≥44 or R(6,6)≥103) would be a genuinely celebrated, decades-standing result with high field prestige; but it is exactly what's walled. Real-world relevance nil. Net: moderate abstract impact, ~zero attainable impact.","verifier":"Given an n-vertex graph G, check G contains no K_k and its complement contains no K_k. Poly-time for fixed k (clique enumeration; trivial in practice for k=5,6,7 up to n~150). Cheap, exact — same verifier class as our R(5,5) work.","attack":"If attempted: SAT/ILP/max-independent-set + tabu/simulated-annealing over circulant (cyclic) 2-colorings, seeking a K_k-free-in-both-colors witness at record+1 vertices. R(5,5) n=43 cyclic space (~2^21) is trivially exhaustible and already empty; the hard part is the full non-cyclic n=43 space (~10^130 graphs), which is the wall. R(6,6) circulant n=103 is ~2^50, borderline-feasible on a 100-core farm (~days) but has been searched for decades with nothing. Rough compute to even re-confirm the R(5,5) cyclic wall: hours; to make real progress: infeasible / needs a new construction idea, not compute.","our_edge":"None beyond generic compute. This is literally our R(5,5) machinery, but we already closed R(5,5) as no-cheap-witness, and the neighboring diagonal records are decades-hardened against precisely our SAT/local-search farm. No structural asymmetry or lemma-based pruning edge here the way the R(5,5) rigidity certs gave us.","verdict":"PARTIAL — headline is a pure asymptotic/analytic conjecture (NONE for the theorem, and the page itself says finite computation can't resolve it). The in-class finite sub-problem (record diagonal clique-free Ramsey-graph lower bounds) is a PERFECT structural fit for our verifier-first witness farm — but every specific frontier is a WALL: R(5,5)≥43 conjectured tight and already witness-hunted by us with nothing; R(6,6)/R(8,8) lowers static for 55–60 years against our exact toolset; large-k records are deterministic Paley graphs; all live progress is asymptotic. Do not attack for a record. Only residual value: R(5,5) is already documented in our stack, so no new work warranted.","lane":"witness-local-search","board_class":"READY","board_class_reason":"witness = 2-coloring of K43; mono-K5 count, exact, cheap (our R(5,5) verifier); a 0-defect coloring proves R(5,5) >= 44 - EinsteinArena-style market board","wall_reason":null,"erdos_url":"https://www.erdosproblems.com/1029","links":{"oeis":[],"arxiv":[]},"frontier":{"summary":"R(5,5) in [43,46]; best public K43 colorings have exactly 2 monochromatic K5s (a 0-defect witness proves R(5,5) >= 44)"}},{"id":1135,"title":"Collatz (3n+1) convergence","prize":"$500","statement":"Erdős #1135 ($500): with f(n)=n/2 (even), f(n)=(3n+1)/2 (odd), does every m≥1 reach 1 under iteration of f? This is the Collatz conjecture.","finite_object":"Two candidate finite witnesses: (a) a disproof — a nontrivial cycle (finite orbit of integers, exactly checkable) or the certified start of a provably divergent orbit; (b) a verification extension — certifying every m ≤ N reaches 1, which extends OEIS A008908/path-record tables. Neither settles the (infinite, uncomputable) headline; the positive claim is explicitly NOT finite-checkable.","current_record":"Verification: all m ≤ 2^71 ≈ 2.36×10^21 reach 1 — David Barina, published Jan 2025 (J. Supercomputing, DOI 10.1007/s11227-025-07337-0), extending his own 2^68 (2020/2021). Method: hyper-optimized GPU sieve (1,335× over naive CPU) distributed across thousands of workers on several European supercomputers. Cycle nonexistence: no nontrivial Collatz m-cycle with m ≤ 91 (Hercher 2022, J. Integer Seq. 2023), building on Simons–de Weger; combined with the 2^69+ verified range this rules out short cycles up to astronomical magnitude.","beatable":"WALL","beatable_reason":"Both finite fronts are walls for our machinery. (1) Counterexample search: massively exhausted — no cycle or divergence below ~2.36×10^21, and cycle-length theory (m≥92) forbids short cycles up to astronomical sizes. A witness almost certainly does not exist in any reachable range; this is not a movable record, it is a near-certain no-witness situation (worse than R(5,5), where the witness at least plausibly existed). (2) Verification-bound extension: the record is held by *exactly* the exhaustion tool-class we would use, but on GPU supercomputer hardware with a 1,335× optimized sieve — a ~100-core CPU farm is 2–4 orders of magnitude behind Barina's multi-supercomputer GPU effort, so we cannot push 2^71. Crucially our differentiated tools (SAT/ILP/max-independent-set/DRAT) give ZERO leverage here: Collatz iteration is a straight arithmetic sieve, not a constraint-satisfaction structure — there is no clause encoding, no independent-set reduction, no UNSAT certificate to produce. And even a successful extension claims no prize and is a trivial table increment, not a contribution.","fit_score":2,"impact_score":3,"impact_reason":"Collatz is one of the most famous open problems, so nominal fame is high, but the prize is only $500 and — decisively — is UNCLAIMABLE by finite computation (the problem file itself states it cannot be resolved by a finite computation). The only finite contribution we could even attempt (nudging a verification bound) is a low-significance table increment already done better by others, with zero real-world relevance. Achievable impact for us is near zero.","verifier":"Trivial and exact: iterate f from a seed until it hits 1 (converged) or exceeds the tested bound; a cycle witness is verified by following the finite orbit back to its start (O(cycle length) exact integer arithmetic). Verification cost is O(1) memory per seed but the *search* is O(N) seeds — the checker is cheap, the exhaustion is the cost.","attack":"No viable attack. If forced: a residue-class sieve (2^k-block reachability with precomputed acceleration tables) on the CPU farm could re-verify up to ~2^60–2^63 in days, but that is far below the 2^71 frontier — pure catch-up, no new record. SAT/ILP/MIS are inapplicable. The only \"novel\" finite angle (search for a nontrivial cycle via the cycle-structure Diophantine constraints of Simons–de Weger/Hercher) is already pushed past m=91 by dedicated number-theoretic bounding, not brute search, and is not a fit for our solvers.","our_edge":"None beyond generic compute — and our compute is strictly inferior to the incumbent's GPU-supercomputer sieve. Our SAT/DRAT/MIS moat does not apply to an arithmetic iteration problem.","verdict":"PARTIAL headline / WALL for us. Per the rubric this is PARTIAL — the asymptotic conjecture is NONE (uncomputable, no finite witness for the positive claim), but a genuine finite sub-problem exists (counterexample cycle/divergence search; verification-bound extension). However, both sub-problems are WALLS for our specific machinery: the counterexample is near-certainly nonexistent in reach and exhaustively searched; the verification frontier (2^71, Barina Jan 2025) is held by a superior GPU-supercomputer sieve we cannot out-compute with a 100-core CPU farm; and our differentiating tools (SAT/ILP/MIS/DRAT) provide no leverage on straight-line arithmetic iteration. DO NOT PURSUE — no exact-witness edge, unclaimable prize, no fit to the verifier-first farm.","lane":"wall","board_class":"NONE","board_class_reason":"exhaustion claim (2^71 sieve) has no certificate - adjudication = full recompute at GPU-supercomputer scale; cycle-witness side is an existence bounty with no frontier","wall_reason":"exhaustion claim (2^71 sieve) has no certificate - adjudication = full recompute at GPU-supercomputer scale; cycle-witness side is an existence bounty with no frontier","erdos_url":"https://www.erdosproblems.com/1135","links":{"oeis":[],"arxiv":[]}}]}