{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:erdos-131-non-dividing-sets", "title": "Erdős Problem #131", "canonical_statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any nonempty subset of $A\\setminus\\{a\\}$. Estimate $F(N)$. The lower bound $F(N)\\gg N^{1/5}$ is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave $F(N)\\leq N^{1/4+o(1)}$. The claimed new result is the matching upper bound $F(N)\\leq N^{1/5+o(1)}$, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence $F(N)=N^{1/5+o(1)}$.", "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "candidate", "solution_events": [ { "id": "vibemathed:erdos-131-non-dividing-sets:event", "problem_id": "vibemathed:erdos-131-non-dividing-sets", "type": "proved", "status": "candidate", "occurred_at": "2026-07-24T00:00:00.000Z", "title": "A projective approach to non-dividing sets", "summary": "The new content is the upper bound; the matching N^(1/5) construction is prior work of Erdős and Csaba. erdosproblems.com has not accepted the claim", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:erdos-131-non-dividing-sets:attempt" ], "method_family_ids": [ "vibemathed:erdos-131-non-dividing-sets:method" ], "source_assertion_ids": [ "vibemathed:erdos-131-non-dividing-sets:assertion" ] } ], "attempts": [ { "id": "vibemathed:erdos-131-non-dividing-sets:attempt", "problem_id": "vibemathed:erdos-131-non-dividing-sets", "model": "GPT-5.6 Sol, Claude", "provider": "OpenAI, Anthropic", "attempted_at": "2026-07-24T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "The paper states that the novel idea - the projective normalization that survives the divisibility constraints and drops the associated convex geometry by one dimension, moving the exponent from 1/4 to 1/5 - was found by GPT-5.6 Sol. The Lean formalization was then completed by a Claude agent loop working autonomously against a human-written route document until the development compiled with no sorry and a clean axiom audit.", "failed_routes": [], "artifacts": [ { "label": "Lean 4 formalization", "url": "https://github.com/theofilxeff/erdos_131", "kind": "lean-proof", "license": null }, { "label": "erdosproblems.com/131", "url": "https://www.erdosproblems.com/131", "kind": "problem-record", "license": null } ], "sources": [ { "label": "A projective approach to non-dividing sets", "url": "https://theofilxeff.github.io/Erdos_131.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Erdős Problem #131", "url": "https://vibemathed.com/problem/erdos-131-non-dividing-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:erdos-131-non-dividing-sets:verification", "problem_id": "vibemathed:erdos-131-non-dividing-sets", "solution_event_id": "vibemathed:erdos-131-non-dividing-sets:event", "level": "lean_verified_statement_audited", "mathematical_correctness": "supported", "statement_fidelity": "audited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Built and audited by the site on 2026-08-02. A clean clone of the author's Lean 4 development (50 files, 17,408 lines) compiles against the pinned mathlib revision on Lean 4.32.0 with no sorry, admit or native_decide. `#print axioms Nondividing.main_log_limit` returns exactly the eleven whitelisted axioms - propext, Classical.choice, Quot.sound and the eight declared external interfaces - and notably no sorryAx, so no placeholder is load-bearing. Statement fidelity checked against the trusted Challenge.lean: the definitions of non-dividing and F, and the theorem type log F(N)/log N -> 1/5, match. NOT verified: the eight external axioms are assumed rather than proved. Each cites a published result (Schneider, Rogers-Shephard, Betke-Henk-Wills, Pham-Zakharov Lemmas 1, 7 and 13, Conlon-Fox-Pham) but none was checked line by line against its source, and the density-increment exponent in convex_density_set is where the 1/4 to 1/5 improvement lives. erdosproblems.com still lists the problem open with no comments.", "sources": [ { "label": "VibeMathed: Erdős Problem #131", "url": "https://vibemathed.com/problem/erdos-131-non-dividing-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "A projective approach to non-dividing sets", "url": "https://theofilxeff.github.io/Erdos_131.pdf", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "A projective approach to non-dividing sets", "url": "https://theofilxeff.github.io/Erdos_131.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Erdős Problem #131", "url": "https://vibemathed.com/problem/erdos-131-non-dividing-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:erdos-131-non-dividing-sets:signal:verify_now", "problem_id": "vibemathed:erdos-131-non-dividing-sets", "kind": "verify_now", "reason": "The source labels this result as a candidate; verification should precede reuse.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "The source status is candidate and must not be represented as solved.", "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }