A construction of almost Steiner systems

TL;DR

This paper proves the existence of almost Steiner systems for large n, where each t-set of vertices is covered by one or two edges, advancing the understanding of Steiner system constructions in combinatorics.

Contribution

It demonstrates that for all sufficiently large n, almost Steiner systems with parameters t, k, and n exist, filling a gap in the existence problem for t ≥ 6.

Findings

Existence of almost Steiner systems for large n

Coverage of each t-set by one or two edges

Progress towards nontrivial Steiner systems for t ≥ 6

Abstract

Let $n$, $k$, and $t$ be integers satisfying $n>k>t\ge2$. A Steiner system with parameters $t$, $k$, and $n$ is a $k$-uniform hypergraph on $n$ vertices in which every set of $t$ distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for $t\geq6$. In this note we prove that for every $k>t\ge2$ and sufficiently large $n$, there exists an almost Steiner system with parameters $t$, $k$, and $n$; that is, there exists a $k$-uniform hypergraph on $n$ vertices such that every set of $t$ distinct vertices is covered by either one or two edges.