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

Abstract

For a partially ordered set(A, ≤), letGA be the simple, undirected graph with vertex set A such that two vertices a ≠ ∈ b A are adjacent if either a ≤ b or b a ≤ . We call GA the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G = GA. For a class of simple, undirected graphs and n, m ≥ 1, we define the Ramsey number (n m, ) with respect to to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in consisting of r vertices contains either a complete n-clique Kn 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.