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Erdős Problem #1026: Monotonic Subsequence Sums

Combinatorics · posed by Paul Erdős, 1975 · solved

1 attempt · 1 machine check

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.

Context

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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 attemptAristotle, with GPT, Gemini and AlphaEvolve also contributing with Boris Alexeev, Stijn Cambie, Terence Tao, Lawrence Wu ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    Aristotle, with GPT, Gemini and AlphaEvolve also contributing
    people
    Boris Alexeev, Stijn Cambie, Terence Tao, Lawrence Wu

    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.

    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

      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.

      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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