Rainbow H-factors of complete s-uniform r-partite hypergraphs

We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V-1, V-2, ..., V-r} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k vertices in each vertex class by T-s,T-r(k). In this paper we prove that if h, r and s are positive integers with 2 <= s <= r <= h then there exists a constant k = k(h, r, s) so that if H is an s-uniform hypergraph with h vertices and chromatic number X(H) = r then any proper edge coloring of T-s,T-r(k) has a rainbow H-factor.