{"paper":{"title":"On the Tur\\'an density of $\\{1, 3\\}$-Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Shuliang Bai","submitted_at":"2018-02-16T22:02:22Z","abstract_excerpt":"In this paper, we consider the Tur\\'an problems on $\\{1,3\\}$-hypergraphs. We prove that a $\\{1, 3\\}$-hypergraph is degenerate if and only if it's $H^{\\{1, 3\\}}_5$-colorable, where $H^{\\{1, 3\\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\\{\\{2\\}, \\{3\\}, \\{1, 2, 4\\}, \\{1, 3, 5\\}, \\{1, 4, 5\\}\\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\\{1, 2\\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Tur\\'an densities of some small $\\{1,3\\}$-hypergraphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.06143","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"}