{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:20:13Z","timestamp":1758824413660},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,9,1]],"date-time":"2009-09-01T00:00:00Z","timestamp":1251763200000},"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":[[2009,9]]},"abstract":"<jats:p>For <jats:italic>k<\/jats:italic>-uniform hypergraphs <jats:italic>F<\/jats:italic> and <jats:italic>H<\/jats:italic> and an integer <jats:italic>r<\/jats:italic>, let <jats:italic>c<jats:sub>r,F<\/jats:sub><\/jats:italic>(<jats:italic>H<\/jats:italic>) denote the number of <jats:italic>r<\/jats:italic>-colourings of the set of hyperedges of <jats:italic>H<\/jats:italic> with no monochromatic copy of <jats:italic>F<\/jats:italic>, and let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548309009948_inline1\"><jats:alt-text>$c_{r,F}(n)=\\max_{H\\in\\ccHn} c_{r,F}(H)$<\/jats:alt-text><\/jats:inline-graphic>, where the maximum runs over all <jats:italic>k<\/jats:italic>-uniform hypergraphs on <jats:italic>n<\/jats:italic> vertices. Moreover, let ex(<jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>) be the usual <jats:italic>extremal<\/jats:italic> or <jats:italic>Tur\u00e1n function<\/jats:italic>, <jats:italic>i.e.<\/jats:italic>, the maximum number of hyperedges of an <jats:italic>n<\/jats:italic>-vertex <jats:italic>k<\/jats:italic>-uniform hypergraph which contains no copy of <jats:italic>F<\/jats:italic>.<\/jats:p><jats:p>For complete graphs <jats:italic>F<\/jats:italic> = <jats:italic>K<\/jats:italic><jats:sub>\u2113<\/jats:sub> and <jats:italic>r<\/jats:italic> = 2, Erd\u0151s and Rothschild conjectured that <jats:italic>c<\/jats:italic><jats:sub>2,<jats:italic>K<\/jats:italic><jats:sub>\u2113<\/jats:sub><\/jats:sub>(<jats:italic>n<\/jats:italic>) = 2<jats:sup>ex(<jats:italic>n<\/jats:italic>,<jats:italic>K<\/jats:italic><jats:sub>\u2113<\/jats:sub>)<\/jats:sup>. This conjecture was proved by Yuster for \u2113 = 3 and by Alon, Balogh, Keevash and Sudakov for arbitrary \u2113. In this paper, we consider the question for hypergraphs and show that, in the special case when <jats:italic>F<\/jats:italic> is the Fano plane and <jats:italic>r<\/jats:italic> = 2 or 3, then <jats:italic>c<jats:sub>r,F<\/jats:sub><\/jats:italic>(<jats:italic>n<\/jats:italic>) = <jats:italic>r<\/jats:italic><jats:sup>ex(<jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>)<\/jats:sup>, while <jats:italic>c<jats:sub>r,F<\/jats:sub><\/jats:italic>(<jats:italic>n<\/jats:italic>) \u226b <jats:italic>r<\/jats:italic><jats:sup>ex(<jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>)<\/jats:sup> for <jats:italic>r<\/jats:italic> \u2265 4.<\/jats:p>","DOI":"10.1017\/s0963548309009948","type":"journal-article","created":{"date-parts":[[2009,6,1]],"date-time":"2009-06-01T09:19:20Z","timestamp":1243847960000},"page":"803-818","source":"Crossref","is-referenced-by-count":21,"title":["On Colourings of Hypergraphs Without Monochromatic Fano Planes"],"prefix":"10.1017","volume":"18","author":[{"given":"HANNO","family":"LEFMANN","sequence":"first","affiliation":[]},{"given":"YURY","family":"PERSON","sequence":"additional","affiliation":[]},{"given":"VOJT\u011aCH","family":"R\u00d6DL","sequence":"additional","affiliation":[]},{"given":"MATHIAS","family":"SCHACHT","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2009,9,1]]},"reference":[{"key":"S0963548309009948_ref13","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1985-15403-5"},{"key":"S0963548309009948_ref14","first-page":"637","article-title":"Kl+1-free graphs: Asymptotic structure and a 0\u20131 law","volume":"303","author":"Kolaitis","year":"1987","journal-title":"Trans. Amer. Math. Soc."},{"key":"S0963548309009948_ref19","first-page":"399","volume-title":"Probl\u00e8mes Combinatoires et Th\u00e9orie des Graphes","author":"Szemer\u00e9di","year":"1978"},{"key":"S0963548309009948_ref1","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610704005563"},{"key":"S0963548309009948_ref20","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199604)21:4<441::AID-JGT10>3.0.CO;2-I"},{"key":"S0963548309009948_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2004.12.004"},{"key":"S0963548309009948_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-005-0034-2"},{"key":"S0963548309009948_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20242"},{"key":"S0963548309009948_ref12","unstructured":"[12] Kohayakawa Y. , Nagle B. , R\u00f6dl V. and Schacht M. Weak hypergraph regularity and linear hypergraphs. J. Combin. Theory, Sec. 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PhD thesis, Forschungsinstitut f\u00fcr Diskrete Mathematik, Rheinische Friedrichs\u2013Wilhelms\u2013Universit\u00e4t Bonn."},{"key":"S0963548309009948_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00280-6"},{"key":"S0963548309009948_ref17","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973068.25"},{"key":"S0963548309009948_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788085"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548309009948","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T21:58:38Z","timestamp":1554415118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548309009948\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":20,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["S0963548309009948"],"URL":"https:\/\/doi.org\/10.1017\/s0963548309009948","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9]]}}}