{"paper":{"title":"Separating hypergraph Tur\\'an densities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Sch\\\"ulke, Haotian Yang, Hong Liu, Shuaichao Wang, Yixiao Zhang","submitted_at":"2024-10-11T15:48:18Z","abstract_excerpt":"Determining the Tur\\'an densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Tur\\'an posed this problem in 1941, $\\pi(K_{\\ell}^{(k)})$ remains unknown for all $\\ell>k\\geq 3$. Prior to this work, it was not even known whether $\\pi(K_{\\ell}^{(k)})<\\pi(K_{\\ell+1}^{(k)})$ holds for general $\\ell$ and $k$, and the best-known bounds on $\\pi(K_{\\ell}^{(k)})$ are far from implying anything close to this. We prove that $\\pi(K_{\\ell}^{(k)})<\\pi(K_{\\ell+1}^{(k)})$, for all $\\ell>k\\geq 3$, and provide a general criterion to distinguish the Tur\\'an densities of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.08921","kind":"arxiv","version":2},"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/2410.08921/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"}