{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T12:35:27Z","timestamp":1774528527997,"version":"3.50.1"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"5-6","license":[{"start":{"date-parts":[[2003,12,3]],"date-time":"2003-12-03T00:00:00Z","timestamp":1070409600000},"content-version":"unspecified","delay-in-days":32,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2003,11]]},"abstract":"<jats:p>For a graph <jats:italic>H<\/jats:italic> and an integer <jats:italic>n<\/jats:italic>, the Tur\u00e1n number <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline1.png\"\/> is the maximum possible number of edges in a simple graph on <jats:italic>n<\/jats:italic> vertices that contains no copy of <jats:italic>H<\/jats:italic>. <jats:italic>H<\/jats:italic> is <jats:italic>r<\/jats:italic>-degenerate if every one of its subgraphs contains a vertex of degree at most <jats:italic>r<\/jats:italic>. We prove that, for any fixed bipartite graph <jats:italic>H<\/jats:italic> in which all degrees in one colour class are at most <jats:italic>r<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline2.png\"\/>. This is tight for all values of <jats:italic>r<\/jats:italic> and can also be derived from an earlier result of F\u00fcredi. We also show that there is an absolute positive constant <jats:italic>c<\/jats:italic> such that, for every fixed bipartite <jats:italic>r<\/jats:italic>-degenerate graph <jats:italic>H<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline3.png\"\/> This is motivated by a conjecture of Erd\u0151s that asserts that, for every such <jats:italic>H<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline4.png\"\/><\/jats:p><jats:p>For two graphs <jats:italic>G<\/jats:italic> and <jats:italic>H<\/jats:italic>, the Ramsey number <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline5.png\"\/> is the minimum number <jats:italic>n<\/jats:italic> such that, in any colouring of the edges of the complete graph on <jats:italic>n<\/jats:italic> vertices by red and blue, there is either a red copy of <jats:italic>G<\/jats:italic> or a blue copy of <jats:italic>H<\/jats:italic>. Erd\u0151s conjectured that there is an absolute constant <jats:italic>c<\/jats:italic> such that, for any graph <jats:italic>G<\/jats:italic> with <jats:italic>m<\/jats:italic> edges, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline6.png\"\/>. Here we prove this conjecture for bipartite graphs <jats:italic>G<\/jats:italic>, and prove that for general graphs <jats:italic>G<\/jats:italic> with <jats:italic>m<\/jats:italic> edges, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0963548303005741_inline7.png\"\/> for some absolute positive constant <jats:italic>c<\/jats:italic>.<\/jats:p><jats:p>These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including R\u00f6dl, Kostochka, Gowers and Sudakov.<\/jats:p>","DOI":"10.1017\/s0963548303005741","type":"journal-article","created":{"date-parts":[[2003,12,3]],"date-time":"2003-12-03T14:23:14Z","timestamp":1070461394000},"page":"477-494","source":"Crossref","is-referenced-by-count":98,"title":["Tur\u00e1n Numbers of Bipartite Graphs and Related Ramsey-Type Questions"],"prefix":"10.1017","volume":"12","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Krivelevich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benny","family":"Sudakov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2003,12,3]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548303005741","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T09:22:45Z","timestamp":1589534565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548303005741\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,11]]},"references-count":0,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2003,11]]}},"alternative-id":["S0963548303005741"],"URL":"https:\/\/doi.org\/10.1017\/s0963548303005741","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,11]]}}}