{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,6]],"date-time":"2024-04-06T00:39:14Z","timestamp":1712363954251},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T00:00:00Z","timestamp":1701216000000},"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>Given a fixed graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline1.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and a constant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline2.png\" \/><jats:tex-math>\n$c \\in [0,1]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we can ask what graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline3.png\" \/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with edge density <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline4.png\" \/><jats:tex-math>\n$c$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> asymptotically maximise the homomorphism density of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline5.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline6.png\" \/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. For all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline7.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline8.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the maximising <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline9.png\" \/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline10.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and densities <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline11.png\" \/><jats:tex-math>\n$c$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that the optimising graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline12.png\" \/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is neither the quasi-star nor the quasi-clique (Day and Sarkar, <jats:italic>SIAM J. Discrete Math.<\/jats:italic> 35(1), 294\u2013306, 2021). We also show that for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline13.png\" \/><jats:tex-math>\n$c$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> large enough all graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline14.png\" \/><jats:tex-math>\n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> maximise on the quasi-clique (Gerbner et al., <jats:italic>J. Graph Theory<\/jats:italic> 96(1), 34\u201343, 2021), and for any <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline15.png\" \/><jats:tex-math>\n$c \\in [0,1]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the density of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline16.png\" \/><jats:tex-math>\n$K_{1,2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is always maximised on either the quasi-star or the quasi-clique (Ahlswede and Katona, <jats:italic>Acta Math. Hung.<\/jats:italic> 32(1\u20132), 97\u2013120, 1978). Finally, we extend our results to uniform hypergraphs.<\/jats:p>","DOI":"10.1017\/s096354832300041x","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T02:56:23Z","timestamp":1701226583000},"page":"300-318","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Threshold graphs maximise homomorphism densities"],"prefix":"10.1017","volume":"33","author":[{"given":"Grigoriy","family":"Blekherman","sequence":"first","affiliation":[]},{"given":"Shyamal","family":"Patel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,29]]},"reference":[{"key":"S096354832300041X_ref5","doi-asserted-by":"publisher","DOI":"10.1080\/15427951.2008.10129166"},{"key":"S096354832300041X_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(85)90080-9"},{"key":"S096354832300041X_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(94)90388-3"},{"key":"S096354832300041X_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/19M1296525"},{"key":"S096354832300041X_ref11","volume-title":"Threshold Graphs and Related Topics","author":"Mahadev","year":"1995"},{"key":"S096354832300041X_ref3","volume-title":"The Probabilistic Method","author":"Alon","year":"2004"},{"key":"S096354832300041X_ref13","first-page":"238","article-title":"Maximum star densities","volume":"55","author":"Reiher","year":"2018","journal-title":"Stud. Sci. Math. Hung."},{"key":"S096354832300041X_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22563"},{"key":"S096354832300041X_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-017-1804-0"},{"key":"S096354832300041X_ref10","volume-title":"Large Networks and Graph Limits","volume":"60","author":"Lov\u00e1sz","year":"2012"},{"key":"S096354832300041X_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000389"},{"key":"S096354832300041X_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01902206"},{"key":"S096354832300041X_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2011.03.009"},{"key":"S096354832300041X_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761855"},{"key":"S096354832300041X_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771528"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832300041X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T10:18:33Z","timestamp":1712312313000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832300041X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,29]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["S096354832300041X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832300041x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,29]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. 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"}]}}