Asymmetric Ramsey numbers of trees

TL;DR

This paper determines the exact asymmetric Ramsey numbers for large classes of trees under certain degree and bipartition constraints, using new bounds and constructions.

Contribution

It establishes conditions under which the asymmetric Ramsey number for pairs of trees equals a new lower bound, advancing understanding of tree Ramsey numbers.

Findings

Exact Ramsey numbers for large classes of trees are determined.

The lower bound matches the actual Ramsey number under specified conditions.

Counterexamples show the bounds do not hold if assumptions are relaxed.

Abstract

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.