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Erdős Problem #43

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erdos-43Number theoryposed by 1982recorded: disproved

1 attempt · 1 machine check · no person has looked

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|?

Context

both proposed bounds fail

A numbered Erdos problem that carries a $100 Erdos prize, setting it above the typical entry in the catalog.

People

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Interest

nobody looking

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.5 Pro, Aristotle, Claude ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.5 Pro, Aristotle, Claude

    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

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed site check ·

      scope Reproduction by the VibeMathed site

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

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    Endorsements

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    Discussion of this attempt

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Discussion

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