Ramsey numbers of books and quasirandomness

The book graph B (k) n consists of n copies of Kk+1 joined along a common Kk. The Ramsey numbers of B (k) n 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 Erd˝os, 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 Szemer´edi’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