Consider a ground set of $n$ elements. I am interested in the following question. What is the largest number $m$ (in terms of $n$) such that there exists a collection of $m$ sets $S_1,S_2,\ldots,S_m$ such that for all $i$:
$$ S_i \setminus(\cup_{j \neq i}S_j) \neq \emptyset. $$
I wonder if there is an explicit answer. If not, is there any non-trivial large $m$ such that such a system is guaranteed to exist?