Ramsey numbers of digraphs with local edge structure

Abstract

One of the classical topics in graph Ramsey theory is the study of which $n$-vertex graphs have Ramsey numbers that are linear in $n$. In this paper, we consider this problem in the context of directed graphs. The oriented Ramsey number of a digraph $G$ is the smallest integer $N$ such that every $N$-vertex tournament contains a copy of $G$. We prove that every bounded-degree acyclic digraph with a ``local edge structure'' has a linear oriented Ramsey number. More precisely, we say that a digraph $G$ has graded bandwidth $w$ if its vertices can be partitioned into sets $V_1, \dots, V_H$ such that all edges $uv \in E(G)$ with $u \in V_i$ and $v \in V_j$ satisfy $1 \leq j - i \leq w$. We prove that $\vec{r}(G) \leq 3^{57\Delta w} |V(G)|$ for any acyclic $G$ with graded bandwidth $w$ and maximum degree $\Delta$. This provides a common generalization of several prior results, including on digraphs of bounded height, of digraphs of bounded bandwidth, and blowups of bounded-degree oriented trees. This notion also captures a wide variety of natural digraphs, such as oriented grids and hypercubes.