{"paper":{"title":"Tur\\'an's Theorem for random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Bobby DeMarco, Jeff Kahn","submitted_at":"2015-01-07T00:46:17Z","abstract_excerpt":"For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \\geq b_r(G)$ for any $G$, and Tur\\'an's Theorem says that equality holds for complete graphs. With $G_{n,p}$ the usual (\"binomial\" or \"Erd\\H{o}s-R\\'enyi\") random graph, we show:\n  For each fixed r there is a C such that if \\[ p=p(n) > Cn^{-\\tfrac{2}{r+1}}\\log^{\\tfrac{2}{(r+1)(r-2)}}n, \\] then $\\Pr(t_r(G_{n,p})=b_r(G_{n,p}))\\rightarrow 1$ as $n\\rightarrow\\infty$.\n  This is best possible (apart from the value of $C$) and settles a question first conside"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01340","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"}