Ramsey numbers of partial order graphs ...

For a partially ordered set (A, ≤) , let G_A 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 G_A 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 = G_A . For a class of simple, undirected graphs and n, m ≥ 1 , we define the Ramsey number R(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 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.