It has been proved by Moon and Moser in 1965 that any finite simple graph has at most $3^{|V|/3}$ maximal cliques. Still, some hereditary classes of graphs have very few maximal cliques in comparison to the general case: for example 4-hole free odd-signable graphs have at most $|V|+2|E|$ maximal cliques.

Is it known a similar (not to say as tight as) upper-bound on the number of maximal cliques for other hereditary classes of graphs ? In particular, is it already known an improved (w.r.t. $3^{|V|/3}$) upper bound for perfect graphs, claw-free graphs or odd-hole free graphs ?