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Erdős Problem #131

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erdos-131-non-dividing-setsNumber theoryposed by Paul Erdős, 1975recorded: candidate

1 attempt · 1 machine check · no person has looked

Statement

Let F(N)F(N) be the maximal size of A⊆{1,…,N}A\subseteq\{1,\ldots,N\} such that no a∈Aa\in A divides the sum of any nonempty subset of A∖{a}A\setminus\{a\}. Estimate F(N)F(N). The lower bound F(N)≫N1/5F(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)≤N1/4+o(1)F(N)\leq N^{1/4+o(1)}. The claimed new result is the matching upper bound F(N)≤N1/5+o(1)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)=N1/5+o(1)F(N)=N^{1/5+o(1)}.

Context

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

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Sol, Claude ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol, Claude

    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.

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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