{"paper":{"title":"Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guanghui Wang, Hao Lin, Wenling Zhou","submitted_at":"2026-07-26T16:49:30Z","abstract_excerpt":"A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \\emph{$(n,d,\\mu)$-dense} if $e_H(X,Y,Z)\\ge d|X||Y||Z|-\\mu n^3$ for all $X,Y,Z\\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness.\n  In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,\\mu)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $\\delta_1(H)$ and the minimum codegree $\\delta_2(H)$. We actually prove a general result: define \\[\n  f(d):=\\frac{1-\\sqrt{(4d-1)/3}}2. \\] We prove that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23748","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/2607.23748/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"}