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Pach's Tangency Conjecture: Improved Bounds

Combinatorics · posed by János Pach · partial

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Statement

Pach conjectured that nn Jordan arcs, pairwise crossing exactly once with no triple points, have O(n)O(n) tangent pairs. The best known bound stood at O(n7/4)O(n^{7/4}); the paper improves it to O(n3/2)O(n^{3/2}) (and O(n5/3)O(n^{5/3}) in the at-most-one-crossing relaxation), plus a tight Θ(n4/3)\Theta(n^{4/3}) for a grounded x-monotone variant.

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

  1. proof attempt · #1

    Eyal Ackerman and Balázs Keszegh, using Gemini

    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 assisted
    — a person led the work and used a model along the way.

    "For the proof of Theorem 9 we used some back and forth interaction with Google's Large Language Model Gemini." One theorem of the paper, attributed plainly.

    Exponent improvements toward Pach's conjecture, which remains open.

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