{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T03:23:52Z","timestamp":1777433032730,"version":"3.51.4"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T00:00:00Z","timestamp":1700092800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The well-known Erd\u0151s-Hajnal conjecture states that for any graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline1.png\"\/><jats:tex-math>\n$F$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, there exists <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline2.png\"\/><jats:tex-math>\n$\\epsilon \\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline3.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-vertex graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline4.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> that contains no induced copy of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline5.png\"\/><jats:tex-math>\n$F$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has a homogeneous set of size at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline6.png\"\/><jats:tex-math>\n$n^{\\epsilon }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We consider a variant of the Erd\u0151s-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline7.png\"\/><jats:tex-math>\n$m$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline8.png\"\/><jats:tex-math>\n$f$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges for any positive <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline9.png\"\/><jats:tex-math>\n$m$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline10.png\"\/><jats:tex-math>\n$0\\leq f \\leq \\binom{m}{2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline11.png\"\/><jats:tex-math>\n$m=4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, for every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline12.png\"\/><jats:tex-math>\n$S \\subseteq \\{0,1,2,3,4\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline13.png\"\/><jats:tex-math>\n$S$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In most cases the bounds are essentially tight. We also determine, for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline14.png\"\/><jats:tex-math>\n$S$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.<\/jats:p>","DOI":"10.1017\/s0963548323000433","type":"journal-article","created":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T05:52:53Z","timestamp":1700113973000},"page":"286-299","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Large cliques or cocliques in hypergraphs with forbidden order-size pairs"],"prefix":"10.1017","volume":"33","author":[{"given":"Maria","family":"Axenovich","sequence":"first","affiliation":[]},{"given":"Domagoj","family":"Brada\u010d","sequence":"additional","affiliation":[]},{"given":"Lior","family":"Gishboliner","sequence":"additional","affiliation":[]},{"given":"Dhruv","family":"Mubayi","sequence":"additional","affiliation":[]},{"given":"Lea","family":"Weber","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,16]]},"reference":[{"key":"S0963548323000433_ref3","doi-asserted-by":"publisher","DOI":"10.1137\/05064357X"},{"key":"S0963548323000433_ref21","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/s004930200012","article-title":"Large cliques in \n\n\n\n$C_4$\n\n\n-free graphs","volume":"22","author":"Gy\u00e1rf\u00e1s","year":"2002","journal-title":"Combinatorica"},{"key":"S0963548323000433_ref4","doi-asserted-by":"crossref","unstructured":"[4] Baker, R. 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Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}