Asymmetric Ramsey numbers of trees

Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number $R(T,S)$ is lower bounded by $\underline{R}(T,S)=\max\{n+\tau_2,\nu+\min\{t_2,\nu\},\min\{2t_1,2\nu\},2\tau_1\}-1$. Our main result shows that there exists a constant $c>0$, such that for all sufficiently large integers $n\geq\nu$, if (i) $\Delta(T)\leq cn/\log n$ and $\Delta(S)\leq c\nu/\log\nu$, (ii) $\tau_2\geq t_2$, and (iii) $\nu\geq t_1$, then $R(T,S)=\underline{R}(T,S)$. In particular, this determines the exact Ramsey numbers for a large family of pairs of trees. We also provide examples showing that $R(T,S)$ can exceed $\underline{R}(T,S)$ if any one of the three assumptions (i), (ii), and (iii) is removed.