  Let $HP_{n,m,k}$ 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 $m=Kn\log n$ 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 $m=Kn\log n$ where $K$ is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)}_{n,m}$. Here
$G^{(n)}_{n,m}$ denotes a random edge coloring of $G_{n,m}$ with $n$ colors.
When $n$ is odd, our proof requires $m=\om(n\log n)$ for there to be a rainbow
Hamilton cycle.
