A construction of almost Steiner systems

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.