Affiliation:
1. Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, Sharjah, United Arab Emirates
2. Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee, Austria
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.