Ramsey Numbers of Books and Quasirandomness

David Conlon*, Jacob Fox, Yuval Wigderson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The book graphBn(k) consists of n copies of Kk+1 joined along a common Kk. The Ramsey numbers of Bn(k) are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi’s regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom.

Original languageEnglish
Pages (from-to)309-363
Number of pages55
Issue number3
StatePublished - Jun 2022
Externally publishedYes


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