For a partially ordered set (A,\le ) , let {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a\ne b\in A are adjacent if either a\le b or b\le 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 {\mathcal{C}} of simple, undirected graphs and n, m\ge 1 , we define the Ramsey number { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in {\mathcal{C}} 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.