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Erdős Problem #138

Combinatorics · posed by 1981 · solved

1 attempt · 1 machine check

Statement

If W(k)W(k) is the least NN such that every two-colouring of {1,…,N}\{1, \dots, N\} contains a monochromatic kk-term arithmetic progression, must W(k+1)−W(k)→∞W(k+1) - W(k) \to \infty?

Context

the stronger question W(k)1/k→∞W(k)^{1/k} \to \infty remains open

A numbered Erdos problem that carries a $500 Erdos prize and an unusually dense reference trail on erdosproblems.com (15 sources), setting it above the typical entry in the catalog.

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 attemptAlphaProof Nexus ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    AlphaProof Nexus

    Proved by AlphaProof Nexus with a Lean-checked argument.

    the stronger question W(k)1/k→∞W(k)^{1/k} \to \infty remains open

    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

      Lean-checked; official Erdős problems record updated.

      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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