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 {V1, V2, . . . , Vr} 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 Ts,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 χ(H) = r then any proper edge coloring of Ts,r(k) has a rainbow H-factor.