Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]
Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]
This is trivially true if $\mathcal{F}$ does not contain any bipartite graphs, since by the Erdős-Stone theorem if $H\in\mathcal{F}$ has minimal chromatic number $r\geq 2$ then \[\mathrm{ex}(n;H)=\mathrm{ex}(n;\mathcal{F})=\left(\frac{r-2}{r-1}+o(1)\right)\binom{n}{2}.\] Erdős and Simonovits observe that this is false for infinite families $\mathcal{F}$, e.g. the family of all cycles.
See also [575] and the entry in the graphs problem collection.