Is it true that, if $\epsilon>0$ and $n$ is sufficiently large, whenever $m\leq (2-\epsilon)2^{n/2}$ the graph $G_\mathcal{F}$ has $<2^{n}$ many edges?
Is it true that if $G_{\mathcal{F}}$ has $\geq cm^2$ edges then $m\ll_c 2^{n/2}$?
Is it true that, for any $\epsilon>0$, there exists some $\delta>0$ such that if there are $>m^{2-\delta}$ edges then $m<(2+\epsilon)^{n/2}$?
For the first question we need to take $\epsilon>0$ since since if $n$ is even and $m=2^{n/2+1}$ one could take $\mathcal{F}$ to be all subsets of $\{1,\ldots,n/2\}$ together with $\{1,\ldots,n/2\}$ union all subsets of $\{n/2+1,\ldots,n\}$, which produces $2^{n}$ edges.
The third question was answered in the affirmative by Alon and Frankl [AlFr85], who proved that, for every $k\geq 1$, if $m=2^{(\frac{1}{k+1}+\delta)n}$ for some $\delta>0$ then the number of edges is \[< \left(1-\frac{1}{k}\right)\binom{m}{2}+O(m^{2-\Omega_k(\delta^{k+1})}).\] They also answer the second question in the negative, noting that if $\mathcal{F}$ is the family of sets which either intersect $\{n/2+1,\ldots,n\}$ in at most $1$ element or intersect $\{1,\ldots,n/2\}$ in at least $n/2-1$ elements then $m \gg n2^{n/2}$ and there are at least $2^{-5}\binom{m}{2}$ edges.
Finally, an affirmative answer to the first question follows from Theorem 1.4 and Corollary 1.5 of Alon, Das, Glebov, and Sudakov [ADGS15].
