Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory

Abstract

For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or \emph{comparability graph} of $A$. Further, we say that a graph $G$ is a partial order graph if there exists a partially ordered set $A$ such that $G = G_A$. For a class $\mathcal{C}$ of simple, undirected graphs and $n$, $m \ge 1$, we define the Ramsey number $\mathcal{R}_{\mathcal{C}}(m,n)$ with respect to $\mathcal{C}$ to be the minimal number of vertices $r$ such that every induced subgraph of an arbitrary partial order graph consisting of $r$ vertices contains either a complete $n$-clique $K_n$ or an independent set consisting of $m$ vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.