An Erdos-Ko-Rado theorem for direct products

Let n(i), k(i) be positive integers, i=1,..., d, satisfying n(i) greater than or equal to 2k(i). Let X(1),...,X(d) be pairwise disjoint sets with \X(i)\ = n(i). Let H be the family of those (k(1) +...+ k(d))-element sets which have exactly k(i) elements in X(i), i = 1,..., d. It is shown that F subset of H is an intersecting family then \H\/\H\less than or equal to max(i)k(i)/n(i), and this is best possible. The proof is algebraic, although in the d = 2 case a combinatorial argument is presented as well.