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.

Threshold Ramsey multiplicity for paths and even cycles

“…We remark that there is a strengthening of the regularity lemma, proved in [6], where each part is -regular with all but an -fraction of the other parts and each part is also -regular with itself. Working with this variant rather than Lemma 7 would allow us to simplify our proof very slightly.…”

Ramsey numbers of large books

“…We will frequently use the following consequence, which can be used to count extensions of cliques and thus estimate the size of books, see [[8], Corollary 2.6].…”

“…A refined version of the regularity lemma [Conlon [7], Lemma 3] guarantees that one can find a regular subset in each part of the partition for any graph. A key ingredient of the proof of our main result is the following refined regularity lemma [Conlon, Fox and Wigderson [8], Lemma 2.1], which is a slight strengthening of that due to Conlon [7] and the usual version of Szemerédi's regularity lemma [24]. An earlier refined version of the regularity lemma due to Alon, Fischer, Krivelevich and Szegedy [1] has been used in the proof of the induced removal lemma.…”

“…Wigderson [8]. For more Ramsey numbers of large books versus other graphs, we refer the reader to [12,13,15,18,23] and other related references.…”

“…Conlon [7] proved that $$r({B}_{n}^{(k)},{B}_{n}^{(k)})=({2}^{k}+{o}_{k}(1))n$$, which confirms a conjecture of Thomason [25] asymptotically and also gives an answer to a problem proposed by Erdős [9]. Using a different method, the upper bound has been improved to $$r({B}_{n}^{(k)},{B}_{n}^{(k)})\le {2}^{k}n+{O}_{k}\left(\frac{n}{{(\mathrm{logloglog}n)}^{1\unicode{x02215}25}}\right)$$ by Conlon, Fox and Wigderson [8]. For more Ramsey numbers of large books versus other graphs, we refer the reader to [12,13,15,18,23] and other related references.…”