An Erdos-Ko-Rado Theorem for Direct Products

Let n i , k i be positive integers , i 5 1 , . . . , d , satisfying n i > 2 k i . Let X 1 , . . . , X d be pairwise disjoint sets with u X i u 5 n i . Let * be the family of those ( k 1 1 ? ? ? 1 k d )-element sets which have exactly k i elements in X i , i 5 1 , . . . , d . It is shown that if ̂ ’ * is an intersecting family then u ̂ u / u * u < max i k i / n i , and this is best possible . The proof is algebraic , although in the d 5 2 case a combinatorial argument is presented as well .