{"paper":{"title":"Stability results for graphs with a critical edge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Roberts, Alex Scott","submitted_at":"2016-10-26T15:59:24Z","abstract_excerpt":"The classical stability theorem of Erd\\H{o}s and Simonovits states that, for any fixed graph with chromatic number $k+1 \\ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible number of edges can be made into the $k$-partite Tur\\'{a}n graph by adding and deleting $o(n^2)$ edges. In this paper, we prove sharper quantitative results for graphs $H$ with a critical edge, both for the Erd\\H{o}s-Simonovits Theorem (distance to the Tur\\'{a}n graph) and for the closely related question of how close an $H$-free graph is to being $k$-partite. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08389","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"}