Ramsey Goodness of Bounded Degree Trees

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue colouring 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) ≥ (|G|-1)(χ(H)-1)+σ(H), where χ(H) is the chromatic number of H and σ(H) is the size of the smallest colour class in a χ(H)-colouring of H. A graph G is called H-good if R(G, H) = (|G|-1)(χ(H)-1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdos in 1983 and has been extensively studied since then. In this paper we show that if n ≥ Ω(|H| log⁴ |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 Erdos, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|⁴).