Ramsey numbers of bounded degree trees versus general graphs

Abstract

For every $k\ge 2$ and Δ, we prove that there exists a constant $C_{\Delta ,k}$ such that the following holds. For every graph H with $\chi \left(H\right)=k$ and every tree T with at least $C_{\Delta ,k}\left|H\right|$ vertices and maximum degree at most Δ, the Ramsey number $R(T,H)$ is $(k-1)\left(\right|T|-1)+\sigma \left(H\right)$, where $\sigma \left(H\right)$ is the size of a smallest colour class across all proper k-colourings of H. This is tight up to the value of $C_{\Delta ,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.