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Erdős Problem #670: Diameter with Separated Distances

Combinatorics · posed by Paul Erdős · disproved

1 attempt · 1 machine check

Statement

Erdős asked whether every nn-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least (1+o(1))n2(1+o(1))n^2. Disproved: an explicit high-dimensional construction beats the conjectured constant.

Context

A numbered problem from the Erdős catalog: real and documented, with a specialist audience.

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

    constructionGPT-5.4 Pro, Harmonic Aristotle with Boon Suan Ho ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.4 Pro, Harmonic Aristotle
    people
    Boon Suan Ho

    "GPT-5.4 Pro was used to discover the construction of this paper, and Harmonic Aristotle was used to formalize the proof in Lean 4, with some assistance from GPT-5.4 Pro." All arguments independently verified by the author.

    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: partially checked

      Recorded from Lean ·

      scope Lean formalization of the core argument; statement correspondence not independently audited

      The proof is formalized in Lean 4 by Harmonic Aristotle; the formalization is public. Tier: the formalization is by Harmonic Aristotle with author verification only - and as of August 2026, erdosproblems.com still lists #670 as OPEN, so the canonical tracker has not yet accepted the disproof.

      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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