Ramsey numbers of books and quasirandomness

TL;DR

This paper simplifies the proof of the asymptotic order of Ramsey numbers for book graphs, avoids using Szemerédi's regularity lemma for tighter bounds, and proves that extremal colorings are quasirandom.

Contribution

It provides a simpler proof of known asymptotic results, offers a new proof avoiding Szemerédi's regularity lemma, and confirms that extremal colorings are quasirandom.

Findings

Simplified proof of asymptotic order of Ramsey numbers for book graphs.

A new proof that avoids Szemerédi's regularity lemma, tightening error bounds.

Proof that all extremal colorings are quasirandom.

Abstract

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 Erd\H{o}s, 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.