{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T11:21:10Z","timestamp":1648898470581},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T00:00:00Z","timestamp":1624924800000},"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":[[2022,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is well known that for any integers <jats:italic>k<\/jats:italic> and <jats:italic>g<\/jats:italic>, there is a graph with chromatic number at least <jats:italic>k<\/jats:italic> and girth at least <jats:italic>g<\/jats:italic>. In 1960s, Erd\u00f6s and Hajnal conjectured that for any <jats:italic>k<\/jats:italic> and <jats:italic>g<\/jats:italic>, there exists a number <jats:italic>h<\/jats:italic>(<jats:italic>k<\/jats:italic>,<jats:italic>g<\/jats:italic>), such that every graph with chromatic number at least <jats:italic>h<\/jats:italic>(<jats:italic>k<\/jats:italic>,<jats:italic>g<\/jats:italic>) contains a subgraph with chromatic number at least <jats:italic>k<\/jats:italic> and girth at least <jats:italic>g<\/jats:italic>. In 1977, R\u00f6dl proved the case when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000250_inline1.png\" \/><jats:tex-math>\n$g=4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, for arbitrary <jats:italic>k<\/jats:italic>. We prove the fractional chromatic number version of R\u00f6dl\u2019s result.<\/jats:p>","DOI":"10.1017\/s0963548321000250","type":"journal-article","created":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T06:17:26Z","timestamp":1624947446000},"page":"136-143","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Triangle-free subgraphs with large fractional chromatic number"],"prefix":"10.1017","volume":"31","author":[{"given":"Bojan","family":"Mohar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hehui","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,6,29]]},"reference":[{"key":"S0963548321000250_ref13","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22571"},{"key":"S0963548321000250_ref7","author":"Erd\u00f6s","year":"1969"},{"key":"S0963548321000250_ref10","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1002\/rsa.3240070302","article-title":"The Ramsey number R(3,t) has order of magnitude","volume":"7","author":"Kim","year":"1995","journal-title":"Random Struct. Algorithms"},{"key":"S0963548321000250_ref6","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1959-003-9"},{"key":"S0963548321000250_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-004-0010-2"},{"key":"S0963548321000250_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/fms.2016.28"},{"key":"S0963548321000250_ref5","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.jctb.2011.04.002","article-title":"A note on Thomassen\u2019s conjecture","volume":"101","author":"Dellamonica","year":"2011","journal-title":"J. Comb. Theory Ser. B"},{"key":"S0963548321000250_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(78)90022-5"},{"key":"S0963548321000250_ref14","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1995.1004"},{"key":"S0963548321000250_ref1","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/0097-3165(80)90030-8","article-title":"A note on Ramsey numbers","volume":"29","author":"Ajtai","year":"1980","journal-title":"J. Comb. Theory Ser. A"},{"key":"S0963548321000250_ref2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1976-133-5"},{"key":"S0963548321000250_ref3","unstructured":"[3] Bukh, B. Interesting problems. http:\/\/www.borisbukh.org\/problems.html. See also http:\/\/www.openproblemgarden. org\/category\/bukh_boris."},{"key":"S0963548321000250_ref15","doi-asserted-by":"crossref","unstructured":"[15] R\u00f6dl, V. (1977) On the chromatic number of subgraphs of a given graph. Proc. Am. Math. Soc. 64 370\u2013371.","DOI":"10.1090\/S0002-9939-1977-0469806-4"},{"key":"S0963548321000250_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90067-9"},{"key":"S0963548321000250_ref17","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199609)23:1<33::AID-JGT3>3.0.CO;2-L"},{"key":"S0963548321000250_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0163-9"},{"key":"S0963548321000250_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20534"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548321000250","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T10:38:57Z","timestamp":1639996737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548321000250\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,29]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["S0963548321000250"],"URL":"https:\/\/doi.org\/10.1017\/s0963548321000250","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,29]]},"assertion":[{"value":"\u00a9 The Author(s), 2021. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}