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Erdős Problem #138

Combinatorics · posed by 1981 · solved

2 comments

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?

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

  1. proof attempt · #1

    AlphaProof Nexus

    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.

    Proved by AlphaProof Nexus with a Lean-checked argument.

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

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

    lean: correctLean

    scope Lean formalization of the result

    Lean-checked; official Erdős problems record updated.

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

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