Ramsey-type results for unions of comparability graphs

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 rootn. 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 n(1/3). On the other hand, there exist such graphs. for which the size 4 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.