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Erdős's Planar Unit Distance Conjecture

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erdos-planar-unit-distanceGeometry & topologyposed by Paul Erdős, 1946recorded: disproved

1 attempt · 1 machine check · no person has looked

Statement

Conjectured upper bound on how many pairs among nn points in the plane can be exactly one unit apart.

Context

Erdős's 1946 unit-distance problem, a founding question of combinatorial geometry.

People

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Projects

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Interest

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionOpenAI frontier model (specific version not disclosed) with Noga Alon, Thomas Bloom, Timothy Gowers, Daniel Litt, Will Sawin, Jacob Tsimerman, Melanie Matchett Wood ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    OpenAI frontier model (specific version not disclosed)
    people
    Noga Alon, Thomas Bloom, Timothy Gowers, Daniel Litt, Will Sawin, Jacob Tsimerman, Melanie Matchett Wood

    Model-assisted construction of a point configuration with more than n1.014n^{1.014} unit-distance pairs, beating the conjectured bound.

    Reviews

    0 human 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 VibeMathed expert verification (imported) ·

      scope Expert review reported upstream; the reviewer has no probXiv account and is not credited here

      The counterexample was generated by an OpenAI model; the human-verified version was written up by Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman and coauthors, who call it a short, digested, human-verified version of the construction. They attribute the crucial ideas, in retrospect, to Ellenberg-Venkatesh, Golod-Shafarevich and Hajir-Maire-Ramakrishna.

      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.

    Endorsements

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    Discussion of this attempt

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Discussion

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