Anti-Ramsey Colorings in Several Rounds

A t-round χ-coloring is defined as a sequence ψ1, …, ψt of t (not necessarily distinct) edge colorings of a complete graph, using at most χ colors in each of the colorings. For positive integers k⩽n and t let χt(k, n) denote the minimum number χ of colors for which there exists a t-round χ-coloring of Kn such that all (k2) edges of each Kk⊆Kn 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 χt(3, n)=Θ(n1/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 Kn with the same (k2) colors such that all (k2) edges of each Kk⊆Kn 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.