Rainbow \(H\)-factors of complete \(s\)-uniform \(r\)-partite hypergraphs (Q1010683)

Summary: 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 \({\mathcal T}_{s,r}(k)\). In this paper we prove that if \(h,\;r\) and \(s\) are positive integers with \(2\leq s\leq r\leq h\) 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 \(\chi(H)=r\) then any proper edge coloring of \({\mathcal T}_{s,r}(k)\) has a rainbow \(H\)-factor.