Ramsey numbers of bounded degree trees versus general graphs

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