Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $\mathcal{F}$ is a family of at most $c{2}^n$ many finite sets of size $n$. Let $X={\cup }_{A\in \mathcal{F}}A$.Must there exist ${\gg }_c{2}^{|X|}$ many sets $B\subset X$ which intersect every set in $\mathcal{F}$, yet contain none of them?