Ramsey goodness of bounded degree trees

TL;DR

This paper proves that large enough bounded degree trees are Ramsey good with respect to any graph H, with the size threshold nearly optimal up to logarithmic factors, improving previous bounds significantly.

Contribution

It establishes a near tight bound on the size of bounded degree trees needed to be H-good, advancing the understanding of Ramsey properties of trees.

Findings

Bounded degree trees of size at least Omega(|H| log^4 |H|) are H-good.

Improves previous bound from Omega(|H|^4) to near optimal.

The dependency between tree size and |H| is tight up to logarithmic factors.

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 $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, 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 Erd\H{o}s in 1983 and has been extensively studied since then. In this paper we show that if $n\geq \Omega(|H| \log^4 |H|)$ 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 Erd\H{o}s, Faudree, Rousseau, and Schelp from 1985, who proved that $n$-vertex bounded degree trees are $H$-good when when $n \geq \Omega(|H|^4)$.