Universal Cycle Packings and Coverings for k-Subsets of an n-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let (p_{n,k}) be the length of a longest (n, k)-Ucycle packing and (c_{n,k}) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed (k,p_{n,k}={natopwithdelims ()k}-O(n^{lfloor k/2rfloor })). Moreover, when k is not fixed, we prove that if (k=k(n)le n^{alpha }), where (0, then (p_{n,k}={natopwithdelims ()k}-o({natopwithdelims ()k}^beta )) and (c_{n,k}={natopwithdelims ()k}+o({natopwithdelims ()k}^beta )), for some (beta <1). Finally, we show that if (k=o(n)), then (p_{n,k}={natopwithdelims ()k}(1-o(1))).