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Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \max_{0 \le x \le 1}\min\{x^3, 1-x\} \approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.
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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.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Single-author arXiv preprint; not yet peer-reviewed.
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the question is answered negatively and the correct threshold is determined
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.