Let HPn,m,k be drawn uniformly from all k-uniform, k-partite
hypergraphs where each part of the partition is a disjoint copy of [n]. We
let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge
randomly from one of \k colors. We show that if \k=n and m=Knlogn where
K is sufficiently large then w.h.p. there is a rainbow colored perfect
matching. I.e. a perfect matching in which every edge has a different color. We
also show that if n is even and m=Knlogn where K is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in Gn,m(n). Here
Gn,m(n) denotes a random edge coloring of Gn,m with n colors.
When n is odd, our proof requires m=\om(n\log n) for there to be a rainbow
Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph