{"paper":{"title":"Dense $3$-uniform hypergraphs containing a large clique","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Biao Wu, Yuejian Peng","submitted_at":"2017-01-22T08:43:48Z","abstract_excerpt":"An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$ of $G$ satisfies $\\lambda (G') < \\lambda (G)$, where $\\lambda (G)$ is the Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $\\pi(F)$, the Tur\\'an density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all dense $F$-hom-free $r$-uniform hypergraphs. This connection has been applied in estimating Tur\\'an density of hypergraphs. When $r=2$, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when $r\\ge 3$, it b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06139","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":""},"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"}