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.
Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The maximum n
umber of such embeddings is asymptotically determined for all finite posets $P$ as $frac{{n choose lfloor n/2rfloor}}{M(P)}$, where $M(P)$ denotes the minimal size of the convex hull of a copy of $P$. We discuss both weak and strong (induced) embeddings.
