Induced C5-free graphs of fixed density: counting and homogeneous sets

For c ∈ (0, 1) let Pn(c) denote the set of n-vertex perfect graphs with density c and Cn(c) the set of n-vertex graphs without induced C5 and with density c. We show that log2 |Pn(c)|/ ( n 2 ) = log2 |Cn(c)|/ ( n 2 ) = h(c) + o(1) with h(c) = 12 if 1 4 ≤ c ≤ 3 4 and h(c) = 1 2H(|2c− 1|) otherwise, where H is the binary entropy function. Furthermore, we use this result to deduce that almost all graphs in Cn(c) have homogeneous sets of linear size. This answers a special case of a question raised by Loebl et al.