{"paper":{"title":"Exact Tur\\'{a}n densities in triple systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caihong Yang, Hongbin Zhao, Nannan Chen, Yuzhen Qi","submitted_at":"2025-07-10T00:59:12Z","abstract_excerpt":"In this paper, we prove several new Tur\\'{a}n density results for $3$-graphs. We show: $\\pi(C_4^3, \\mathrm{complement\\ of\\ } F_5) = 2\\sqrt{3} - 3$, $\\pi(F_{3,2}, C_5^{3-}) = \\frac{2}{9}$, and $\\pi(F_{3,2}, \\mathrm{induced\\ complement\\ of\\ } F_{3,2}) = \\frac{3}{8}$. The first result confirms the conjecture of Shi~[On Tur\\'an denisties of small triple graphs, European J. Combin. 52 (2016) 95-102]. The other results give several special non-principal family posed by Mubayi and R\\\"odl~[On the Tur\\'an number of triple systems, J. Combin. Theory A. 100 (2002) 135-152]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.07360","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/2507.07360/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"}