A Construction of Almost Steiner Systems

Abstract

Abstract: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. C 2013 Wiley Periodicals, Inc. J. Combin. Designs 22: [488][489][490][491][492][493][494] 2014

The existence of designs

“…They also obtain much more precise estimates than we do for the number of designs (within their range of parameters). Another recent result, due to Ferber, Hod, Krivelevich and Sudakov [6] gives a short probabilistic construction of 'almost Steiner systems', in which every r-subset is covered by either one or two q-subsets.…”

“…We regard cliques as the same if they are identical as a subset of K r n : we do not distinguish multiple edges 6. Sums of (multi)graphs are defined by viewing them as vectors over N.…”

Supersaturation of C4: From Zarankiewicz towards Erdős–Simonovits–Sidorenko

“…We remark in the end of this section that a recent result of Ferber, Hod, Krivelevich and Sudakov [13] proved the existence of certain 2 − (n, k, 1) almost designs. However, their result cannot be applied to our case since they assume that the number of blocks is much larger than the number of points, but it could be applied for F (n, n ′ , m) to determine the minimal number of C 4 s in m-edge subgraphs of very much unbalanced bipartite graphs K n,n ′ (n ≪ n ′ ).…”

Hypergraph Matchings and Designs

“…In the actual algorithm we have to exclude any triangle that uses an edge covered by a previous step of the algorithm, but if we are covering a sparse graph one can show that whp at most half (say) of the choices are forbidden at each step, so any whp estimate in the binomial process will hold in the actual process up to a factor of two. (This idea gives a much simpler proof of the result of [27]. )…”

“…An improved bound on λ and a probabilistic method (a local limit theorem for certain random walks in high dimensions) for constructing many other rigid combinatorial structures was recently given by Kuperberg, Lovett and Peled [75]. Ferber, Hod, Krivelevich and Sudakov [27] gave a construction of 'almost Steiner systems', in which every r-subset is covered by either one or two q-subsets.…”