Near universal cycles for subsets exist

Details Near universal cycles for subsets exist Near universal cycles for subsets exist

2008-09-22

arxiv.org/abs/0809.3725v1

Mathematics

Combinatorics

14 pages, 3 figures

Scientific paper

Let S be a cyclic n-ary sequence. We say that S is a {\it universal cycle} ((n,k)-Ucycle) for k-subsets of [n] if every such subset appears exactly once contiguously in S, and is a Ucycle packing if every such subset appears at most once. Few examples of Ucycles are known to exist, so the relaxation to packings merits investigation. A family {S_n} of (n,k)-Ucycle packings for fixed k is a near-Ucycle if the length of S_n is $(1-o(1))\binom{n}{k}$. In this paper we prove that near-(n,k)-Ucycles exist for all k.

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