Ramsey non-goodness involving books

Abstract

In 1983, Burr and Erdős initiated the study of Ramsey goodness problems. Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, 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 (2023) 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 $n\ge 1$, $1\le a_1\le \cdots \le a_{p-1}\le t$ 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 $a_1=a_2=1$.

We disprove the conjecture of Fox et al. (2023). Building upon the work of Fox et al., we make a substantial step by showing that for every $k,p,t\ge 2$, there exists $\delta >0$ such that the following holds for all large n. Let $1\le a_1\le \dots \le a_{p-1}\le t$ and $a_p\le \delta n$ be positive integers. If $a_1=1$, then $r(H,B_{k,n})\le (p-1)(n-1)+k(p-1)(a_2-1)+1$. The inequality is tight if $a_2\left|\right(n-1-k)$. Moreover, we prove that for every $k,a\ge 1$ and $p\ge 2$, there exists $\delta >0$ such that for all large n and $b\le \delta \ln n$, $r\left(K_p\right(1,a,b,\dots ,b),B_{k,n})=(p-1)(n-1)+k(p-1)(a-1)+1$ if $a\left|\right(n-1-k)$, 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.