Rainbow matchings and Hamilton cycles in random graphs

Let HP_n,m,k be drawn uniformly from all m-edge, k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HP_n,m,k(^κ) be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ=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 G_n,m(ⁿ). Here G_n,m(ⁿ) denotes a random edge coloring of G_n,m with n colors. When n is odd, our proof requires m=ω(nlogn) for there to be a rainbow Hamilton cycle.