{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,20]],"date-time":"2026-07-20T11:57:38Z","timestamp":1784548658773,"version":"3.55.0"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2011,10,12]],"date-time":"2011-10-12T00:00:00Z","timestamp":1318377600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,11]]},"abstract":"<jats:p>The <jats:italic>Tur\u00e1n number<\/jats:italic> of a graph <jats:italic>H<\/jats:italic>, ex(<jats:italic>n, H<\/jats:italic>), is the maximum number of edges in any graph on <jats:italic>n<\/jats:italic> vertices which does not contain <jats:italic>H<\/jats:italic> as a subgraph. Let <jats:italic>P<\/jats:italic><jats:sub><jats:italic>l<\/jats:italic><\/jats:sub> denote a path on <jats:italic>l<\/jats:italic> vertices, and let <jats:italic>k<\/jats:italic> \u22c5 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>l<\/jats:italic><\/jats:sub> denote <jats:italic>k<\/jats:italic> vertex-disjoint copies of <jats:italic>P<\/jats:italic><jats:sub><jats:italic>l<\/jats:italic><\/jats:sub>. We determine ex(<jats:italic>n, k<\/jats:italic> \u22c5 <jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>) for <jats:italic>n<\/jats:italic> appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(<jats:italic>n, k<\/jats:italic> \u22c5 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>l<\/jats:italic><\/jats:sub>) for arbitrary <jats:italic>l<\/jats:italic>, and <jats:italic>n<\/jats:italic> appropriately large relative to <jats:italic>k<\/jats:italic> and <jats:italic>l<\/jats:italic>. We provide some background on the famous Erd\u0151s\u2013S\u00f3s conjecture, and conditional on its truth we determine ex(<jats:italic>n, H<\/jats:italic>) when <jats:italic>H<\/jats:italic> is an equibipartite forest, for appropriately large <jats:italic>n<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0963548311000460","type":"journal-article","created":{"date-parts":[[2011,10,12]],"date-time":"2011-10-12T10:32:26Z","timestamp":1318415546000},"page":"837-853","source":"Crossref","is-referenced-by-count":54,"title":["Tur\u00e1n Numbers of Multiple Paths and Equibipartite Forests"],"prefix":"10.1017","volume":"20","author":[{"given":"NEAL","family":"BUSHAW","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"NATHAN","family":"KETTLE","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2011,10,12]]},"reference":[{"key":"S0963548311000460_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20261"},{"key":"S0963548311000460_ref13","first-page":"436","article-title":"Egy gr\u00e1felm\u00e9leti sz\u00e9ls\u0151\u00e9rt\u00e9kfeladatr\u00f3l","volume":"48","author":"Tur\u00e1n","year":"1941","journal-title":"Mat. es Fiz. Lapok."},{"key":"S0963548311000460_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20083"},{"key":"S0963548311000460_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02124681"},{"key":"S0963548311000460_ref12","first-page":"279","volume-title":"Theory of Graphs","author":"Simonovits","year":"1968"},{"key":"S0963548311000460_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-010-0999-5"},{"key":"S0963548311000460_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58043-7_12"},{"key":"S0963548311000460_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2007.08.047"},{"key":"S0963548311000460_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02024498"},{"key":"S0963548311000460_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199602)21:2<229::AID-JGT13>3.0.CO;2-E"},{"key":"S0963548311000460_ref7","first-page":"593","article-title":"On maximal paths and cycles in a graph","volume":"18","author":"Kopylov","year":"1977","journal-title":"Soviet Math. Dokl."},{"key":"S0963548311000460_ref3","volume-title":"Modern Graph Theory","author":"Bollob\u00e1s","year":"2002"},{"key":"S0963548311000460_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(95)00207-D"},{"key":"S0963548311000460_ref14","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4064\/cm-3-1-19-30","article-title":"On the theory of graphs","volume":"3","author":"Tur\u00e1n","year":"1954","journal-title":"Colloquium Math."},{"key":"S0963548311000460_ref10","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1997.1758"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000460","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T03:41:02Z","timestamp":1556336462000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000460\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,12]]},"references-count":15,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2011,11]]}},"alternative-id":["S0963548311000460"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000460","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,12]]}}}