{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:11:17Z","timestamp":1758823877917,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>By using the Szemer\u00e9di Regularity Lemma, Alon and Sudakov recently extended the classical Andr\u00e1sfai-Erd\u0151s-S\u00f3s theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any $(r+1)$-partite graph $H$ whose smallest part has $t$ vertices, there exists a constant $C$ such that for any given $\\varepsilon&gt;0$ and sufficiently large $n$ the following is true. Whenever $G$ is an $n$-vertex graph with minimum degree $$\\delta(G)\\geq\\left(1-{3\\over 3r-1}+\\varepsilon\\right)n,$$  either $G$ contains $H$, or we can delete $f(n,H)\\leq Cn^{2-{1\\over t}}$ edges from $G$ to obtain an $r$-partite graph. Further, we are able to determine the correct order of magnitude of $f(n,H)$ in terms of the Zarankiewicz extremal function.<\/jats:p>","DOI":"10.37236\/293","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:17:18Z","timestamp":1578698238000},"source":"Crossref","is-referenced-by-count":8,"title":["Dense $H$-Free Graphs are Almost $(\\chi(H)-1)$-Partite"],"prefix":"10.37236","volume":"17","author":[{"given":"Peter","family":"Allen","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,1,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T19:01:45Z","timestamp":1579287705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,29]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/293","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,1,29]]},"article-number":"R21"}}