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Erdős Problem #654

Geometry & topology · posed by 1987 · variant

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Statement

If nn planar points have no four concyclic, must some point determine (1−o(1))n(1 - o(1))n distinct distances? Failing that, can one always force more than (1/3+c)n(1/3 + c)n?

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

  1. construction · #1

    Aletheia (Gemini Deep Think)

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
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    — the result was found by a model.

    the strongest form is disproved via configurations where every point sees at most about 3n/4 distinct distances; the weaker improvement remains open

  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

    Expert-reviewed within the Aletheia project, with public report and transcripts; no journal review.

    Repeated from the source; nothing was checked here.

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