Ramsey-Type Results for Unions of Comparability Graphs

Abstract.

 It is well known that the comparability graph of any partially ordered set of n elements contains either a clique or an independent set of size at least . In this note we show that any graph of n vertices which is the union of two comparability graphs on the same vertex set, contains either a clique or an independent set of size at least . On the other hand, there exist such graphs for which the size of any clique or independent set is at most n ^0.4118. Similar results are obtained for graphs which are unions of a fixed number k comparability graphs. We also show that the same bounds hold for unions of perfect graphs.