If $\mathcal{F}$ is a family of subsets of $\{1,\dots ,n\}$ then we write $G_{\mathcal{F}}$ for the graph on $\mathcal{F}$ where $A\sim B$ if $A$ and $B$ are comparable - that is, $A\subseteq B$ or vice versa.Is it true that, if $ϵ>0$ and $n$ is sufficiently large, whenever $m\le (2-ϵ)2^{n/2}$ the graph $G_{\mathcal{F}}$ has $<2^n$ many edges?Is it true that if $G_{\mathcal{F}}$ has $\ge c{m}^2$ edges then $m{\ll }_c{2}^{n/2}$?Is it true that, for any $ϵ>0$, there exists some $\delta >0$ such that if there are $>m^{2-\delta }$ edges then $m<(2+ϵ{)}^{n/2}$?