An Erdős–Ko–Rado Theorem for Direct Products

Letni, kibe positive integers,i=1, ..., d,satisfyingni≥2ki.LetX1, ..., Xdbe pairwise disjoint sets with |Xi| =ni.LetHbe the family of those (k1+···+kd)-element sets which have exactlykielements inXi, i=1,..., d.It is shown that ifF⊂His an intersecting family then |F|/|H|≤maxiki/ni,and this is best possible. The proof is algebraic, although in thed=2 case a combinatorial argument is presented as well.