All Random Solved Random Open
Is it true that the number of graphs on $n$ vertices which do not contain $G$ is \[\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?\]
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 \[2^{(1+c)\mathrm{ex}(n;C_6)}\] 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 \[2^{O(\mathrm{ex}(n;G))}\] holds for all $G$.
Additional thanks to: Tuan Tran