Union-Intersecting Set Systems

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 ≤ 4, and asymptotically if s + t ≥ 5. 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.