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Erdős Problem #43

Number theory · posed by 1982 · disproved

2 comments

Statement

If Sidon sets A,B⊆{1,…,N}A, B \subseteq \{1, \dots, N\} satisfy (A−A)∩(B−B)={0}(A-A) \cap (B-B) = \{0\}, must (∣A∣2)+(∣B∣2)≤(f(N)2)+O(1)\binom{|A|}{2} + \binom{|B|}{2} \le \binom{f(N)}{2} + O(1), where f(N)f(N) is the largest Sidon-set size in [N][N] - and can the bound be improved by a fixed proportion when ∣A∣=∣B∣|A| = |B|?

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    GPT-5.5 Pro, Aristotle, Claude

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    The equal-size bound is disproved by an explicit construction; the unrestricted bound fails as a consequence of the resolution of Erdős Problem #42.

    both proposed bounds fail

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    The official Erdős problems record marks both questions answered negatively, with component Lean proofs.

    Repeated from the source; nothing was checked here.

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open in Claude →open in ChatGPT →

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