Solution of a covering problem related to labelled tournaments

Suppose we have a tournament with edges labelled so that the edges incident with any vertex have at most k distinct labels (and no vertex has outdegree 0). Let m be the minimal size of a subset of labels such that for any vertex there exists an outgoing edge labelled by one of the labels in the subset. It was known that m ? k+1 2 for any tournament. We show that this bound is almost best possible, by a probabilistic construction of tournaments with m = (k 2 = log k). We give explicit tournaments with m = k 2?o(1). If the number of vertices is bounded by N < 2 k , we have a better upper bound of m = O(k log N), which is again almost optimal. We also consider a relaxation of this problem in which instead of the size of a subset of labels we minimize the total weight of a fractional set with analogous properties. In that case the optimal bound is 2k ? 1.