Is it true that the number of graphs on $n$ vertices which do not contain $G$ is
If $G$ is not bipartite the answer is yes, proved by Erdős, Frankl, and Rödl [ErFrRo86]
. The answer is no for $G=C_6$, the cycle on 6 vertices. Morris and Saxton [MoSa16]
have proved there are at least
such graphs for infinitely many $n$, for some constant $c>0$. It is still possible (and conjectured by Morris and Saxton) that the weaker bound of
holds for all $G$.