a graph is called $h$-free if it contains no copy of $h$. denote by $f_n(h)$ the number of (labeled) $h$-free graphs on $n$ vertices. since every subgraph of an $h$-free graph is also $h$-free, it immediately follows that $f_n(h) \geq 2^{\mathrm{ex}(n,h)}$. erd{\h o}s conjectured that, provided that $h$ contains a cycle, this trivial lower bound is asymptotically tight, i.e., \[ f_n(h) = 2^{(1+o(1))\mathrm{ex}(n,h)}. \] the conjecture was resolved in the affirmative for graphs with chromatic number at least $3$ by erd{\h o}s, frankl, and r{\"o}dl (1986)