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Araujo-Piga-Schacht Question on Tight Hamilton Cycles

Combinatorics · posed by Araujo, Piga, Schacht · disproved

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Statement

Araujo, Piga and Schacht asked whether density and codegree both above 1/41/4 force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is p0=max⁡0≤x≤1min⁡{x3,1−x}≈0.3177p_0 = \max_{0 \le x \le 1}\min\{x^3, 1-x\} \approx 0.3177, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every p>1/3p > 1/3 the asymptotically sharp minimum-codegree threshold is determined.

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

  1. construction · #1

    Xichao Shu, using ChatGPT

    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
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    The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.

    the question is answered negatively and the correct threshold is determined

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