{"paper":{"title":"The typical structure of graphs with no large cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, J\\'ozsef Balogh, Maryam Sharifzadeh, Maur\\'icio Collares Neto, Neal Bushaw, Robert Morris","submitted_at":"2014-06-26T17:56:49Z","abstract_excerpt":"In 1987, Kolaitis, Pr\\\"omel and Rothschild proved that, for every fixed $r \\in \\mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \\leqslant (\\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erd\\H{o}s-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollob\\'as and Simonovits."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6961","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"}