Asymmetric Ramsey numbers of trees

Let $n\ge \nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\ge t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes ${\tau }_1\ge {\tau }_2$. Using four natural constructions, we show that the Ramsey number $R(T,S)$ is lower bounded by $\underset{‾}{R}(T,S)=\max \{n+{\tau }_2,\nu +\min \{t_2,\nu \},\min \{2t_1,2\nu \},2{\tau }_1\}-1$. Our main result shows that there exists a constant $c>0$, such that for all sufficiently large integers $n\ge \nu$, if (i) $\mathrm{\Delta }(T)\le cn\mathrm{/}\log n$ and $\mathrm{\Delta }(S)\le c\nu \mathrm{/}\log \nu$, (ii) ${\tau }_2\ge t_2$, and (iii) $\nu \ge t_1$, then $R(T,S)=\underset{‾}{R}(T,S)$. 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 $\underset{‾}{R}(T,S)$ if any one of the three assumptions (i), (ii), and (iii) is removed.