{"paper":{"title":"The Clique Density Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher","submitted_at":"2012-12-11T16:14:44Z","abstract_excerpt":"Tur\\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \\geq 2$ every graph on $n$ vertices with more than ${\\tfrac{r-2}{2(r-1)}\\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs.\n  The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\\'{a}sz and Simonovits conjectured in the 1970s that asymptotically the best possible lower bou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2454","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"}