[{"_id":"22165","oa_version":"Preprint","doi":"10.1017/s0963548322000360","mathsc":["05C55","05D10"],"date_updated":"2026-07-14T08:27:40Z","extern":"1","external_id":{"arxiv":["2110.14483"]},"status":"public","date_published":"2023-05-01T00:00:00Z","keyword":["Ramsey theory","book graphs","Ramsey goodness"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-29T10:53:47Z","article_type":"original","type":"journal_article","publisher":"Cambridge University Press","year":"2023","issue":"3","title":"Off-diagonal book Ramsey numbers","publication_status":"published","OA_place":"repository","publication":"Combinatorics, Probability and Computing","intvolume":"        32","abstract":[{"text":"The book graph 𝐵(𝑘)\r\n𝑛 consists of 𝑛 copies of 𝐾𝑘+1 joined along a common 𝐾𝑘. In the prequel to this paper, we studied the diagonal Ramsey number 𝑟⁡(𝐵(𝑘)\r\n𝑛,𝐵(𝑘)\r\n𝑛). Here we consider the natural off-diagonal variant 𝑟⁡(𝐵(𝑘)\r\n𝑐⁢𝑛,𝐵(𝑘)\r\n𝑛) for fixed 𝑐 ∈(0,1]. In this more general setting, we show that an interesting dichotomy emerges: for very small 𝑐, a simple 𝑘-partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being 𝑘-partite, while, for 𝑐 bounded away from 0, 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.\r\n\r\n","lang":"eng"}],"OA_type":"green","quality_controlled":"1","day":"01","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2110.14483"}],"page":"516-545","citation":{"ieee":"D. Conlon, J. Fox, and Y. Wigderson, “Off-diagonal book Ramsey numbers,” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3. Cambridge University Press, pp. 516–545, 2023.","mla":"Conlon, David, et al. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3, Cambridge University Press, 2023, pp. 516–45, doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>.","short":"D. Conlon, J. Fox, Y. Wigderson, Combinatorics, Probability and Computing 32 (2023) 516–545.","chicago":"Conlon, David, Jacob Fox, and Yuval Wigderson. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>. Cambridge University Press, 2023. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>.","ista":"Conlon D, Fox J, Wigderson Y. 2023. Off-diagonal book Ramsey numbers. Combinatorics, Probability and Computing. 32(3), 516–545.","apa":"Conlon, D., Fox, J., &#38; Wigderson, Y. (2023). Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>","ama":"Conlon D, Fox J, Wigderson Y. Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. 2023;32(3):516-545. doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>"},"article_processing_charge":"No","author":[{"full_name":"Conlon, David","first_name":"David","last_name":"Conlon"},{"first_name":"Jacob","last_name":"Fox","full_name":"Fox, Jacob"},{"first_name":"Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","last_name":"Wigderson","full_name":"Wigderson, Yuval"}],"language":[{"iso":"eng"}],"arxiv":1,"oa":1,"publication_identifier":{"eissn":["1469-2163"],"issn":["0963-5483"]},"volume":32,"month":"05","scopus_import":"1"}]