{"paper":{"title":"Supersaturation Problem for Color-Critical Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oleg Pikhurko, Zelealem B. Yilma","submitted_at":"2012-08-21T17:45:29Z","abstract_excerpt":"The \\emph{Tur\\'an function} $\\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and $\\ex(n,F)+q$ edges can have.\n  We determine $h_F(n,q)$ asymptotically when $F$ is \\emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$.\n  Determining the exact value of $h_F(n,q)$ seems rather "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4319","kind":"arxiv","version":3},"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"}