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The Erdos-Sos Pairwise-Sums Problem

Combinatorics · posed by Paul Erdos, Vera T. Sos · solved

1 attempt · 1 machine check

Statement

Let f3(N)f_3(N) be the least size forcing a set A⊆{1,…,N}A \subseteq \{1,\ldots,N\} to contain distinct a,b,ca,b,c with a+ba+b, a+ca+c and b+cb+c all in AA. The upper bound f3(N)≤5N/8+O(1)f_3(N) \le 5N/8 + O(1) matches the standard construction [N/8,N/4]∪[N/2,N][N/8,N/4] \cup [N/2,N], so f3(N)=5N/8+O(1)f_3(N) = 5N/8 + O(1).

Context

An Erdos-Sos problem on pairwise sums with a standing construction and a gap that this closes exactly.

People

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Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  • #1

    Attempt 1

    proof attemptGPT-5.5 Pro, Aristotle with Ricky Cipollini ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.5 Pro, Aristotle
    people
    Ricky Cipollini

    The paper states that the manuscript was written by GPT-5.5 Pro from a proof developed by the author together with GPT-5.5 Pro, and that the accompanying Lean formalization was carried out with Aristotle. Both the mathematics and the write-up are joint with the model rather than checked by it.

    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

      The paper reports a Lean formalization against Mathlib with no sorries and no added axioms. We have not compiled it. arXiv preprint, not peer-reviewed.

      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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