Near-Universal Cycles for Subsets Exist

Let $S$ be a cyclic $n$-ary sequence. We say that $S$ is a 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 $k$.