Perfect graphs of fixed density: counting and homogeneous sets

For c ∈ (0,1) let C n(c) denote the set of C n-vertex perfect graphs with density c and let C n(c) denote the set of n-vertex graphs without induced C 5 and with density c. We show that lim n→∞ log 2 |Pmathcal;n(c)|/(n 2) = lim n→∞ log 2|C n(c)|/(n 2 = h(c) with h(c) = 1/2 if 1/4 ≤ c ≤ 3/4 and h(c) = 1/2H(|2c - 1|)otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in n(c) have homogeneous sets of linear size. This answers a question raised by Loebl and co-workers.