Perfect Graphs of Fixed Density: Counting and Homogeneous Sets

In this paper we investigate classes of graphs that are defined by forbidding certain substructures. Let H be such a class. We focus on two related goals: to approximate the cardinality of H and to determine the structure of a typical graph in H. In particular, we add the additional constraint that all graphs in H must have the same density c and would like to know how the answer to these questions depends on the parameter c. The quantity |Hn|, where Hn := {G ∈ H : V (G) = [n]}, is also called the speed of H. Often exact formulas or good estimates for |Hn| are out of reach. In these cases, however, one might