Asymmetric Ramsey numbers of trees

TL;DR

This work advances the understanding of asymmetric Ramsey numbers for trees by proving that, under natural sublinear maximum-degree conditions and a refined ordering of bipartition sizes, the asymmetric Ramsey number R(T,S) attains the sharp value $\underline{R}(T,S)=\max\{n+τ_2,ν+\min\{t_2,ν\},\min\{2t_1,2ν\},2τ_1\}-1$. The authors develop a two-pronged approach: a stability analysis using a colored regularity framework and a robust extremal part that shows the extremal configurations indeed force the lower-bound bound to be tight. They introduce a four-stage stability process and four types of extremal structures, then demonstrate embeddings or extremality in each case via five embedding methods built on regularity and weight-function techniques. The paper also constructs counterexamples to show that removing key assumptions (τ_2≥t_2 and ν≥t_1) can make R(T,S) exceed the bound by arbitrary amounts, clarifying the necessity of these hypotheses. Overall, the results yield exact Ramsey numbers for a broad family of tree pairs and deepen the link between sparse graphs, regularity methods, and asymmetric Ramsey phenomena with trees. The findings have significance for combinatorial design and sparse-graph Ramsey theory, offering a blueprint for determining exact values in new asymmetric configurations while identifying precise limits of the current methods.

Abstract

Let $n\geqν$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $ν$-vertex tree with bipartition class sizes $τ_1\geqτ_2$. Using four natural constructions, we show that the Ramsey number $R(T,S)$ is lower bounded by $\underline{R}(T,S)=\max\{n+τ_2,ν+\min\{t_2,ν\},\min\{2t_1,2ν\},2τ_1\}-1$. Our main result shows that there exists a constant $c>0$, such that for all sufficiently large integers $n\geqν$, if (i) $Δ(T)\leq cn/\log n$ and $Δ(S)\leq cν/\logν$, (ii) $τ_2\geq t_2$, and (iii) $ν\geq t_1$, 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.

Figures (3)

• Figure 1: The lower bound constructions \ref{['low:1']}--\ref{['low:4']}.
• Figure 2: The reduced graph structures used in the embedding methods HŁ T and EMa-d.
• Figure 3: An illustration of one potential route in the 4-stage process we use to prove Theorem \ref{['thm:stability2']}.

Theorems & Definitions (81)

• Theorem 1.1: MPY
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 2.1: Azuma's Inequality W
• Lemma 2.2: Chernoff Bound JLR
• Lemma 2.3: Hall's Theorem H