Anti-Ramsey Colorings in Several Rounds

(joint work with Aart Blokhuis, András Gyárfás and Miklós Ruszinkó) For positive integers k ≤ n and t let χ t (k, n) denote the minimum number of colors such that at least in one of the total t colorings of edges of K n all k 2 edges of every K k ⊆ K n get different colors. Generalizing a result of Körner and Simonyi, it is shown in this paper that χ t (3, n) = Θ(n 1/t). Also two-round colorings in cases k > 3 are investigated. Tight bounds for χ 2 (k, n) for all values of k except for k = 5 are obtained. Conversely, let t(k, n) denote the minimum number of colorings such that – having the same k 2 colors in each coloring – at least in one of the total t colorings of K n all k 2 edges of every K k ⊆ K n get different colors. It is also shown, that for k = n/2 t(k, n) is exponentially large. Several related questions are investigated.