Abstract. Let G be a connected graph on n vertices. We prove that if S is a collection of k vertices in G, then there are certainly two vertices u,v in S such that their distance is at most (2n-2)/k. Moreover we prove that, subdivided k-star with rays of lenghts differing by at most 1, is one of the extremal graph. We need this result in a subsequent paper about the maximal Wiener index in a graph with prescribed number of blocks, but we think that it is interesting also on its own.