Abstract
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\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.
Highlights
The Ramsey number R(n, m) gives the solution to the party problem, which asks for the minimum number R(n, m) of guests that must be invited so that at least n will know each other or at least m will not know each other
In the language of graph theory, the Ramsey number is the minimum number of vertices v = R(n, m) such that all undirected simple graphs of order v contain a clique of order n or an independent set of order m
We say that a graph G is a partial order graph if there exists a partially ordered set A such that G = GA
Summary
The Ramsey number R(n, m) gives the solution to the party problem, which asks for the minimum number R(n, m) of guests that must be invited so that at least n will know each other or at least m will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices v = R(n, m) such that all undirected simple graphs of order v contain a clique of order n or an independent set of order m. For a class C of simple, undirected graphs and n, m ≥ 1, we define the Ramsey number RC(n, m) with respect to the class C to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C consisting of r vertices contains either a complete n-clique Kn or an independent set consisting of m vertices.
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