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)|/(n2)=log2|Cn(c)|/(n2)=h(c)+o(1) with h(c)=12 if 14⩽c⩽34 and h(c)=12H(|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.