Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $\mathcal{F}$ is a family of at most $c2^n$ many finite sets of size $n$. Let $X=\cup_{A\in \mathcal{F}}A$.

Must there exist $\gg_c 2^{\lvert X\rvert}$ many sets $B\subset X$ which intersect every set in $\mathcal{F}$, yet contain none of them?