On the Independence Number of Steiner Systems

A partial Steiner (n, r, l)-system is an r-uniform hypergraph on n vertices in which every set of l vertices is contained in at most one edge. A partial Steiner (n, r, l)-system is complete if every set of l vertices is contained in exactly one edge. In a hypergraph H, the independence number α(H) denotes the maximum size of a set of vertices in H containing no edge. In this article we prove the following. Given integers r, l such that r ≥ 2l− 1 ≥ 3, we prove that there exist a partial Steiner (n, r, l)-system H such that α(H) . ( l − 1 r − 1 (r)l ) 1 r−1 n r−l r−1 (log n) 1 r−1 as n→∞. This improves earlier results of Phelps and Rödl, and Rödl and Ŝinajová. We conjecture that it is best possible as it matches the independence number of a random r-uniform hypergraph of the same density. If l = 2 or l = 3, then for infinitely many r the partial Steiner systems constructed are complete for infinitely many n.