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Pach's Tangency Conjecture: Improved Bounds

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pach-tangencies-progressCombinatoricsposed by János Pachrecorded: partial

1 attempt · no person has looked

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.

Context

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

A named Pach conjecture in combinatorial geometry with prior partial results and a documented bound ladder.

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  • #1

    Attempt 1

    proof attemptGemini with Eyal Ackerman, Balázs Keszegh ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    Gemini
    people
    Eyal Ackerman, Balázs Keszegh

    "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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