The book graph \(B_n^{(k)}\) consists of n copies of K_k+1 joined along a common K_k. The Ramsey numbers of \(B_n^{(k)}\) 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.

图书图 \ (B_n^{(k)}\)由沿公共K _k连接的n个K _{k +1}副本组成。已知\(B_n^{(k)}\)的拉姆齐数与经典的集团拉姆齐数有很强的联系。最近，第一作者确定了这些 Ramsey 数的渐近顺序，用于固定k，从而回答了 Erdős、Faudree、Rousseau 和 Schhelp 的一个老问题。在本文中，我们首先提供这个定理的更简单的证明。接下来，回答第一作者的一个问题，我们提出了一个不同的证明，它避免了使用 Szemerédi 的正则引理，从而对错误项提供了更严格的控制。最后，我们通过证明这个 Ramsey 问题的所有极值着色都是拟随机的，证明了 Nikiforov、Rousseau 和 Schhelp 的猜想。