{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T05:25:00Z","timestamp":1740461100129,"version":"3.37.3"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T00:00:00Z","timestamp":1728518400000},"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":[[2025,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For given positive integers <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline1.png\"\/><jats:tex-math>\n$r\\ge 3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline2.png\"\/><jats:tex-math>\n$n$\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=\"S0963548324000269_inline3.png\"\/><jats:tex-math>\n$e\\le \\binom{n}{2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the famous Erd\u0151s\u2013Rademacher problem asks for the minimum number of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline4.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-cliques in a graph with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline5.png\"\/><jats:tex-math>\n$n$\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=\"S0963548324000269_inline6.png\"\/><jats:tex-math>\n$e$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges. A conjecture of Lov\u00e1sz and Simonovits from the 1970s states that, for every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline7.png\"\/><jats:tex-math>\n$r\\ge 3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline8.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is sufficiently large then, for every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline9.png\"\/><jats:tex-math>\n$e\\le \\binom{n}{2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part.<\/jats:p><jats:p>In this note, we explicitly write the minimum number of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline10.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline11.png\"\/><jats:tex-math>\n$r\\ge 4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and all large <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000269_inline12.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, amending the previous conjecture of Pikhurko and Razborov.<\/jats:p>","DOI":"10.1017\/s0963548324000269","type":"journal-article","created":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T12:52:14Z","timestamp":1728564734000},"page":"52-62","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A note on extremal constructions for the Erd\u0151s\u2013Rademacher problem"],"prefix":"10.1017","volume":"34","author":[{"given":"Xizhi","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Oleg","family":"Pikhurko","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2024,10,10]]},"reference":[{"key":"S0963548324000269_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-013-2638-2"},{"key":"S0963548324000269_ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2021.112330"},{"key":"S0963548324000269_ref1","doi-asserted-by":"publisher","DOI":"10.1215\/00192082-10429321"},{"key":"S0963548324000269_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2010.05.013"},{"key":"S0963548324000269_ref18","first-page":"157","article-title":"On a problem of P","volume":"71","author":"Nikiforov","year":"1976\/77","journal-title":"Erd\u0151s. 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