Ramsey goodness of books revisited

TL;DR

This paper provides a new, simpler proof that certain book graphs are Ramsey good for large enough parameters, avoiding complex regularity lemmas and establishing bounds on the size of graphs involved.

Contribution

It introduces a short, regularity-free proof for the Ramsey goodness of book graphs and extends results to more general sparse graphs with explicit bounds.

Findings

Every $B_{k,n}$ with $n \\geq 2^{k^{10p}}$ is $p$-good.

New bounds on Ramsey numbers for specific sparse graphs without using regularity lemmas.

Extended the class of graphs for which tight Ramsey bounds are known.

Abstract

The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n \geq 2^{k^{10p}}$ is $p$-good. Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)| < \delta |V(H)|$ for a small constant $\delta > 0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $\delta n$, for some small constant $\delta>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $\delta$.