{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T08:06:02Z","timestamp":1770710762283,"version":"3.49.0"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T00:00:00Z","timestamp":1673222400000},"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":[[2023,5]]},"abstract":"Abstract<\/jats:title>The book graph$B_n ^{(k)}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>consists of$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>copies of$K_{k+1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>joined along a common$K_k$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In the prequel to this paper, we studied the diagonal Ramsey number$r(B_n ^{(k)}, B_n ^{(k)})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Here we consider the natural off-diagonal variant$r(B_{cn} ^{(k)}, B_n^{(k)})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>for fixed$c \\in (0,1]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In this more general setting, we show that an interesting dichotomy emerges: for very small$c$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, a simple$k$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being$k$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-partite, while, for$c$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>bounded away from$0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of$c$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548322000360","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T10:47:32Z","timestamp":1673261252000},"page":"516-545","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["Off-diagonal book Ramsey numbers"],"prefix":"10.1017","volume":"32","author":[{"given":"David","family":"Conlon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jacob","family":"Fox","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuval","family":"Wigderson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,1,9]]},"reference":[{"key":"S0963548322000360_ref26","doi-asserted-by":"crossref","unstructured":"[26] Zhao, Y. 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Soc."},{"key":"S0963548322000360_ref23","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90273-X"},{"key":"S0963548322000360_ref24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511817106"},{"key":"S0963548322000360_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(74)90133-2"},{"key":"S0963548322000360_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-003-0042-z"},{"key":"S0963548322000360_ref18","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20081"},{"key":"S0963548322000360_ref22","article-title":"Diagonal Ramsey via effective quasirandomness","author":"Sah","journal-title":"Duke Math. J."},{"key":"S0963548322000360_ref14","article-title":"Ramsey goodness of books revisited","author":"Fox","journal-title":"Advances in Combinatorics"},{"key":"S0963548322000360_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(80)90030-8"},{"key":"S0963548322000360_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20973"},{"key":"S0963548322000360_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-021-4409-9"},{"key":"S0963548322000360_ref11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1947-08785-1"},{"key":"S0963548322000360_ref20","first-page":"125","article-title":"The triangle-free process and the Ramsey number $R(3,k)$","volume":"263","author":"Pontiveros","year":"2020","journal-title":"Mem. Am. Math. Soc."},{"key":"S0963548322000360_ref21","first-page":"264","article-title":"On a problem of formal logic","volume":"30","author":"Ramsey","year":"1929","journal-title":"Proc. Lond. Math. Soc."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548322000360","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,4]],"date-time":"2023-12-04T07:24:19Z","timestamp":1701674659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548322000360\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,9]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["S0963548322000360"],"URL":"https:\/\/doi.org\/10.1017\/s0963548322000360","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,9]]},"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"}]}}