{"paper":{"title":"Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xichao Shu","submitted_at":"2026-07-23T17:50:17Z","abstract_excerpt":"We study tight Hamilton cycles in linearly quasirandom 3-graphs. An $n$-vertex 3-graph $H$ is $(p,\\mu)$-dense if $e_H(X,Y,Z)\\ge p|X||Y||Z|-\\mu n^3$ for all $X,Y,Z\\subseteq V(H)$. Ara{\\'u}jo, Piga and Schacht asked whether $p,\\alpha>1/4$ together with $\\delta_2(H)\\ge\\alpha n$ force a tight Hamilton cycle. We give a negative answer: for every $\\varepsilon,\\mu>0$ and all sufficiently large $n$, there exists an $n$-vertex $(p_0-\\varepsilon,\\mu)$-dense 3-graph $H$ with $\\delta_2(H)\\ge(p_0-\\varepsilon)n$ and no tight Hamilton cycle, where $p_0:=\\max_{0\\le x\\le1}\\min\\{x^3,1-x\\}\\approx0.317672$.\n  For"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21568","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.21568/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}