Near universal cycles for subsets exist

TL;DR

This paper proves that near-universal cycles for k-subsets of an n-set exist for all fixed k, providing a near-complete coverage with sequences that are almost as long as the total number of subsets.

Contribution

The authors establish the existence of near-universal cycles for all fixed k, extending the understanding of cycle packings beyond known examples.

Findings

Near-(n,k)-Ucycle packings exist for all fixed k.

Sequences cover asymptotically all k-subsets of [n].

The length of these sequences approaches the total number of k-subsets.

Abstract

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.