Made up, but somewhat interesting:
Let $\lambda\leq\kappa$ be infinite cardinals. Let $X$ be a set of cardinality $\kappa$.
Let $F\subseteq [X]^\kappa$ be a family of $2^\kappa$ subsets, which is closed under taking intersections of $\lambda$-many members.
Let $E\subseteq F$ have the property that for each $f\in F$ there is $e\in E$ with $e\subseteq f$.
How small can $E$ be in general?