{"external_id":{"arxiv":["2109.09205"]},"language":[{"iso":"eng"}],"OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-29T10:58:11Z","oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2109.09205"}],"year":"2023","day":"29","OA_type":"green","doi":"10.19086/aic.2023.4","arxiv":1,"publisher":"Alliance of Diamond Open Access Journals","scopus_import":"1","author":[{"full_name":"Fox, Jacob","last_name":"Fox","first_name":"Jacob"},{"first_name":"Xiaoyu","last_name":"He","full_name":"He, Xiaoyu"},{"last_name":"Wigderson","full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","first_name":"Yuval"}],"date_published":"2023-07-29T00:00:00Z","title":"Ramsey goodness of books revisited","quality_controlled":"1","publication":"Advances in Combinatorics","extern":"1","publication_status":"published","status":"public","_id":"22176","publication_identifier":{"eissn":["2517-5599"]},"type":"journal_article","citation":{"ista":"Fox J, He X, Wigderson Y. 2023. Ramsey goodness of books revisited. Advances in Combinatorics.","short":"J. Fox, X. He, Y. Wigderson, Advances in Combinatorics (2023).","mla":"Fox, Jacob, et al. “Ramsey Goodness of Books Revisited.” <i>Advances in Combinatorics</i>, Alliance of Diamond Open Access Journals, 2023, doi:<a href=\"https://doi.org/10.19086/aic.2023.4\">10.19086/aic.2023.4</a>.","ieee":"J. Fox, X. He, and Y. Wigderson, “Ramsey goodness of books revisited,” <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals, 2023.","chicago":"Fox, Jacob, Xiaoyu He, and Yuval Wigderson. “Ramsey Goodness of Books Revisited.” <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals, 2023. <a href=\"https://doi.org/10.19086/aic.2023.4\">https://doi.org/10.19086/aic.2023.4</a>.","ama":"Fox J, He X, Wigderson Y. Ramsey goodness of books revisited. <i>Advances in Combinatorics</i>. 2023. doi:<a href=\"https://doi.org/10.19086/aic.2023.4\">10.19086/aic.2023.4</a>","apa":"Fox, J., He, X., &#38; Wigderson, Y. (2023). Ramsey goodness of books revisited. <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals. <a href=\"https://doi.org/10.19086/aic.2023.4\">https://doi.org/10.19086/aic.2023.4</a>"},"article_processing_charge":"No","oa_version":"Preprint","article_type":"original","date_updated":"2026-07-14T09:05:44Z","month":"07","abstract":[{"lang":"eng","text":"The Ramsey number r(G,H) is the minimum N such that every graph on N vertices contains G as a subgraph or its complement contains H as a subgraph. For integers n≥k≥1, the k-book Bk,n is the graph on n vertices consisting of a copy of Kk, called the spine, as well as n−k additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph H on n vertices is called p-good if r(Kp,H)=(p−1)(n−1)+1. Nikiforov and Rousseau proved that if n is sufficiently large in terms of p and k, then Bk,n is p-good. Their proof uses Szemerédi's regularity lemma and gives a tower-type bound on n. We give a short new proof that avoids using the regularity method and shows that every Bk,n with n≥2k10p is p-good.\r\nUsing Szemerédi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on r(G,H) for several families of sparse graphs G and H as long as |V(G)|<δ|V(H)| for a small constant δ>0. Using our techniques, we prove a new result of this type, showing that r(G,H)=(p−1)(n−1)+1 when H=Bk,n and G is a complete p-partite graph whose first p−1 parts have constant size and whose last part has size δn, for some small constant δ>0. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on δ.\r\n"}]}