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Erdős Problem #1026: Monotonic Subsequence Sums

Combinatorics · posed by Paul Erdős, 1975 · solved

2 comments

Statement

For a sequence of nn distinct reals, determine the largest constant cc such that some monotonic subsequence always has sum exceeding (c−o(1))⋅(1/n)(c-o(1))\cdot(1/\sqrt{n}) times the total sum. Resolved as c=1c = 1.

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

  1. proof attempt · #1

    Boris Alexeev, Stijn Cambie, Terence Tao and Lawrence Wu, using Aristotle, with GPT, Gemini and AlphaEvolve also contributing

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    Multiple AI systems contributed within an ordinary human mathematical collaboration rather than one model solving it outright; Aristotle produced and formally verified the winning proof in Lean, pinning down the sharp constant c=1c = 1.

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

    lean: correctLean

    scope Lean formalization of the result

    Formally verified in Lean. Documented firsthand by Terence Tao on his blog as a case study in AI-assisted collaboration, not an AI-alone result.

    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 →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.