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Erdős Problem #183: Multicolor Triangle Ramsey

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erdos-183Combinatoricsposed by Paul Erdős, 1961recorded: candidate

1 attempt · 1 machine check · no person has looked

Statement

Let R(3;k)R(3;k) be the least nn such that every kk-colouring of the edges of KnK_n contains a monochromatic triangle. Determine lim⁡k→∞R(3;k)1/k\lim_{k\to\infty} R(3;k)^{1/k} (a $250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is infinite.

Context

A $250 Erdős prize problem from 1961, well known across Ramsey theory.

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionAstra (internal preview) ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Astra (internal preview)

    Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.

    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

      Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, lake build All), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.

      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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