There is an NP-hard problem called Minimum k-Union where we are given a set system with $n$ sets and are asked to select $k$ sets in order to minimize the size of their union.
I'm currently interested in a very similar problem, but don't know how to convert one to another:
Given a set system with $n$ sets and a bound $k$. Select as many sets as possible while their union is at most $k$.
Is this problem NP-hard? Any hint is welcome!