A k-rainbow path in a graph with colored edges is a path of length k where each edge has a different color. In this note, we settle the problem of obtaining a constructive k-coloring of the edges of Kn in which one may find, between any pair of vertices, a large number of internally disjoint k-rainbow paths. In fact, our construction obtains the largest possible number of paths. This problem was considered in a less general setting by Chartrand et al.