The minimum size of k-rainbow connected graphs of given order

Let t(n,k) denote the minimum number of edges of a k-rainbow connected graph G of order n, that is a graph G for which a coloring of the edges with at most k colors exists such that any two vertices of G are connected by a path with edges of pairwise different colors. A general upper bound of t(n,k) and the exact value of t(n,3) are given.