Ramsey numbers of books and quasirandomness

Abstract

The book graph B (k) n consists of n copies of K k+1 joined along a common K k . 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 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.

Three early problems on size Ramsey numbers

“…The following lemma, which shows that colorings which contain good configurations also contain large monochromatic books, was arguably the main idea in [7], but was first stated explicitly in [9]. The statement differs slightly from that in [9,Lemma 3.3], but it is easy to check that the proof carries through. Lemma 3.7.…”

“…This suggests taking the host graph to be a large book, with a spine of order around kn and about 2 k n pages, which is indeed what we do. However, the proof that such a large book is indeed Ramsey for B (k) n is quite involved, requiring substantial input from recent work on Ramsey numbers of books [7,9,10]. This proof is the content of the next subsection.…”

“…To complement the regularity lemma, we have the following consequence of the counting lemma, proved in [9,Corollary 2.6]. It is designed to count monochromatic extensions of cliques and thus estimate the size of monochromatic books.…”

“…Equivalently, it can be described as the join of a K k and an independent set of order n. This K k is called the spine and the vertices of the independent set are called pages. Ramsey numbers of book graphs play a central role in Ramsey theory, because all known techniques for proving upper bounds on the diagonal Ramsey numbers r(K t ) rely on induction schemes that repeatedly use bounds on r(B (k) n ) for appropriately chosen k < t and n. These Ramsey numbers have received considerable attention of late, beginning with work of the first author [7], who asymptotically determined r(B (k) n ) for n sufficiently large in terms of k, and continuing with work of the authors [9,10] giving alternative proofs and exploring variations of the basic question. Regarding size Ramsey numbers, Erdős, Faudree, Rousseau, and Schelp [15] proved that…”

“…In Theorem 1.1, the main new idea is to use a random coloring with a hypergeometric distribution between certain vertices and their higher degree neighbors, rather than the uniform distribution that is usually used in Ramsey-theoretic lower bound constructions. The lower bound in Theorem 1.3 uses a degree-based random coloring, where the probability an edge is red depends on the degrees of its endpoints, while the upper bound uses some of the regularity techniques that were recently developed for studying the ordinary Ramsey numbers of books [7,9,10]. Finally, Theorem 1.4 is proved by examining the properties of an appropriate random graph.…”