Ramsey non-goodness involving books

TL;DR

This paper advances the understanding of Ramsey goodness involving book graphs by establishing new bounds and conjecture verifications without using the regularity lemma, focusing on cases where certain divisibility conditions hold.

Contribution

It proves that the Ramsey number conjecture roughly holds under specific divisibility conditions and provides bounds avoiding the regularity lemma, improving previous results.

Findings

Verified the conjecture when $a_1=a_2=1$.

Established bounds for cases where $a_2$ divides $n-1-k$.

Proved that for large $n$, the Ramsey number equals a specific formula under certain divisibility conditions.

Abstract

In 1983, Burr and Erd\H{o}s initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erd\H{o}s, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let $B_{k,n}$ be the book graph on $n$ vertices which consists of $n-k$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $H=K_p(a_1,\dots,a_{p})$ be the complete $p$-partite graph with parts of sizes $a_1,\dots,a_{p}$. Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2021) revisit several Ramsey goodness results involving books. They comment that it would be very interesting to see how far one can push these ideas. In particular, they conjecture that for all integers $k, p, t\ge 2$, there exists some $\delta>0$ such that for all $n\ge 1$, $1\leq a_1\le\cdots\le a_{p-1}\le t$ and $a_p \le \delta n$, we have $r(H, B_{k,n})= (p-1)(n-1)+d_k(n,K_{a_1,a_2})+1,$ where $d_k(n,K_{a_1,a_2})$ is the maximum $d$ for which there is an $(n+d-1)$-vertex $K_{a_1,a_2}$-free graph in which at most $k-1$ vertices have degree less than $d$.They verify the conjecture when $a_1=a_2=1$. Building upon the work of Fox et al. (2021), we make a substantial step by showing that the conjecture "roughly" holds if $a_1=1$ and $a_2|(n-1-k)$, i.e. $a_2$ divides $n-1-k$. Moreover, avoiding use of the regularity lemma, we prove that for every $k, a\geq 1$ and $p\ge2$, there exists $\delta>0$ such that for all large $n$ and $b\le \delta\ln n$, $r(K_p(1,a,b,\dots,b), B_{k,n})= (p-1)(n-1)+k(p-1)(a-1)+1$ if $a|(n-1-k)$, where the case when $a=1$ has been proved by Nikiforov and Rousseau (2009) using the regularity lemma. The bounds on $1/\delta$ we obtain are not of tower-type since our proofs do not rely on the regularity lemma.