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Erdős Problem #176

Combinatorics · posed by 1965 · partial

2 comments

Statement

Let N(k,ℓ)N(k, \ell) be the least NN such that every f:[N]→{−1,1}f : [N] \to \{-1, 1\} has a kk-term arithmetic progression PP with ∣∑n∈Pf(n)∣≥ℓ|\sum_{n \in P} f(n)| \ge \ell. In particular, is N(k,2)≤CkN(k, 2) \le C^k?

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

  1. proof attempt · #1

    Codex 5.5, ChatGPT-5.5 Pro

    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.

    a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open

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

    lean: correctLean

    scope Lean formalization of the result

    Public Lean proof of the N(k,2) clause.

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

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open in Claude →open in ChatGPT →

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