  Three intersection theorems are proved. First, we determine the size of the
largest set system, where the system of the pairwise unions is l-intersecting.
Then we investigate set systems where the union of any s sets intersect the
union of any t sets. The maximal size of such a set system is determined
exactly if s+t<5, and asymptotically if s+t>4. Finally, we exactly determine
the maximal size of a k-uniform set system that has the above described
(s,t)-union-intersecting property, for large enough n.
