A t-round χ-coloring is defined as a sequence ψ₁, …, ψ_t of t (not necessarily distinct) edge colorings of a complete graph, using at most χ colors in each of the colorings. For positive integers kn and t let χ^t(k, n) denote the minimum number χ of colors for which there exists a t-round χ-coloring of K_n such that all (^k₂) edges of each K_kK_n get different colors in at least one round. Generalizing a result of J. Körner and G. Simonyi (1995, Studia Sci. Math. Hungar.30, 95–103), it is shown in this paper that χ^t(3, n)=Θ(n^1/t). Two-round colorings for k>3 are also investigated. Tight bounds are obtained for χ²(k, n) for all values of k except for k=5. We also study an inverted extremal function, t(k, n), which is the minimum number of rounds needed to color the edges of K_n with the same (^k₂) colors such that all (^k₂) edges of each K_kK_n get different colors in at least one round. For k=n/2, t(k, n) is shown to be exponentially large. Several related questions are investigated. The discussed problems relate to perfect hash functions.