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Erdős's Planar Unit Distance Conjecture

Geometry & topology · posed by Paul Erdős, 1946 · disproved

2 comments

Statement

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

Record

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Noga Alon, Thomas Bloom, Timothy Gowers, Daniel Litt, Will Sawin, Jacob Tsimerman and Melanie Matchett Wood, using OpenAI frontier model (specific version not disclosed)

    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.

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

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed 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.

    Repeated from the source; nothing was checked here.

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