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

For c ∈ ( 0 , 1 ) let P n ( c ) denote the set of n-vertex perfect graphs with density c and C n ( c ) the set of n-vertex graphs without induced C5 and with density c. We show that log 2 | P n ( c ) | / ( n 2 ) = log 2 | C n ( c ) | / ( n 2 ) = h ( c ) + o ( 1 ) with h ( c ) = 1 2 if 1 4 ⩽ c ⩽ 3 4 and h ( c ) = 1 2 H ( | 2 c − 1 | ) otherwise, where H is the binary entropy function. rmore, 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.