{"paper":{"title":"Phase transitions in the Ramsey-Tur\\'an theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Balogh, Mikl\\'os Simonovits, Ping Hu","submitted_at":"2013-04-03T18:01:15Z","abstract_excerpt":"Let $f(n)$ be a function and $L$ be a graph. Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph on $n$ vertices with independence number less than $f(n)$. Erd\\H os and S\\'os asked if $RT\\left(n, K_5, c\\sqrt{n}\\right) = o(n^2)$ for some constant $c$. We answer this question by proving the stronger $RT\\left(n, K_5, o\\left(\\sqrt{n\\log n}\\right)\\right) = o(n^2)$. It is known that $RT \\left(n, K_5, c \\sqrt{n\\log n} \\right) = n^2/4+o(n^2)$ for $c>1$, so one can say that $K_5$ has a Ramsey-Tur\\'an phase transition at $c\\sqrt{n\\log n}$. We extend this result to several other $K_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1036","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":""},"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"}