Ramsey goodness of bounded degree trees

Abstract

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H)\ge (|G|-1)(\chi (H)-1)+\sigma (H)$, where $\chi (H)$ is the chromatic number of $H$ and $\sigma (H)$ is the size of the smallest color class in a $\chi (H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(\chi (H)-1)+\sigma (H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this paper we show that if $n\ge \Omega (|H|{\log }^4|H|)$ then every $n$-vertex bounded degree tree $T$ is $H$-good. The dependency between $n$ and $|H|$ is tight up to $\log$ factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved that $n$-vertex bounded degree trees are $H$-good when when $n\ge \Omega (|H{|}^4)$.