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The Erdos-Sos Pairwise-Sums Problem

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

2 comments

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).

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Ricky Cipollini, using GPT-5.5 Pro, Aristotle

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    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 arXiv preprint, not peer-reviewed.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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