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