A Construction of Almost Steiner Systems

Let n , k , and t be integers satisfying n > k > t ≥ 2 . 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 ≥ 6 . In this note we prove that for every k > t ≥ 2 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.