How to partition sets A(1),…,A(N) in such a way so that elements of each partition are shared between the same number of original sets.

Question

For example, suppose that q(k) denotes number of elements from the original sets shared between k sets (k=1,…,N). Our objective is to find all sets Q such each Q(k) only contains elements which belong to k original sets.

Answer 1

Hint: Let $B$ be the set of all the elements of all the $A$s. Associate each element of $B$ with $k$, the number of sets it is a member of. And then.....