{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:09:37Z","timestamp":1763543377356,"version":"3.45.0"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T00:00:00Z","timestamp":1754870400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,11]]},"abstract":"Abstract<\/jats:title>\n \n The Erd\u0151s\u2013Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erd\u0151s\u2013Simonovits type stability theorem in multi-partite graphs. Different from the Erd\u0151s\u2013Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -free graph\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is close to the extremal graphs for\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , then\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n has a well-defined structure but may be far away from the extremal graphs for\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . As applications, we strengthen a theorem of Bollob\u00e1s, Erd\u0151s, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.\n <\/jats:p>","DOI":"10.1017\/s0963548325100138","type":"journal-article","created":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T05:42:41Z","timestamp":1754890961000},"page":"821-847","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A stability theorem for multi-partite graphs"],"prefix":"10.1017","volume":"34","author":[{"given":"Wanfang","family":"Chen","sequence":"first","affiliation":[{"name":"East China Normal University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changhong","family":"Lu","sequence":"additional","affiliation":[{"name":"East China Normal University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Long-Tu","family":"Yuan","sequence":"additional","affiliation":[{"name":"East China Normal University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,8,11]]},"reference":[{"key":"S0963548325100138_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305007157"},{"key":"S0963548325100138_ref15","doi-asserted-by":"publisher","DOI":"10.4064\/cm-3-1-50-57"},{"key":"S0963548325100138_ref6","first-page":"117","volume-title":"Theory of Graphs","author":"Erd\u0151s","year":"1967"},{"key":"S0963548325100138_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2015.05.001"},{"key":"S0963548325100138_ref11","doi-asserted-by":"publisher","DOI":"10.37236\/10148"},{"key":"S0963548325100138_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000274"},{"key":"S0963548325100138_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01896420"},{"key":"S0963548325100138_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2017.09.016"},{"key":"S0963548325100138_ref7","first-page":"77","volume-title":"Theory of Graphs","author":"Erd\u0151s","year":"1968"},{"key":"S0963548325100138_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548301004758"},{"key":"S0963548325100138_ref19","first-page":"279","volume-title":"Theory of Graphs, Proc. Colloq., Tihany, 1966","author":"Simonovits","year":"1968"},{"key":"S0963548325100138_ref1","first-page":"343","article-title":"Complete subgraphs of chromatic graphs and hypergraphs","volume":"6","author":"Bollob\u00e1s","year":"1974","journal-title":"Utilitas Math."},{"key":"S0963548325100138_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-006-0019-9"},{"key":"S0963548325100138_ref9","first-page":"1089","article-title":"On the structure of linear graphs","volume":"52","author":"Erd\u0151s","year":"1946","journal-title":"Bull. Amer. Math."},{"key":"S0963548325100138_ref22","first-page":"436","article-title":"On an extremal problem in graph theory (in Hungrarian)","volume":"48","author":"Tur\u00e1n","year":"1941","journal-title":"K\u00f6z\u00e9pisk. Mat. \u00e9s Fiz. Lapok"},{"key":"S0963548325100138_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(75)90011-4"},{"key":"S0963548325100138_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2010.05.013"},{"key":"S0963548325100138_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004121000475"},{"key":"S0963548325100138_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-012-2425-5"},{"key":"S0963548325100138_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2016.12.001"},{"key":"S0963548325100138_ref24","first-page":"163","article-title":"On some properties of linear complexes","volume":"24","author":"Zykov","year":"1949","journal-title":"Mat. Sb. N.S."},{"key":"S0963548325100138_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.10.002"},{"volume-title":"Extremal Graph Theory","year":"1978","author":"Bollob\u00e1s","key":"S0963548325100138_ref3"},{"key":"S0963548325100138_ref23","first-page":"19","article-title":"Problem p. 101","volume":"3","author":"Zarakiewicz","year":"1954","journal-title":"Colloq. Math"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100138","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:04:47Z","timestamp":1763543087000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100138\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,11]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["S0963548325100138"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100138","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2025,8,11]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}