On the Independence Number of Steiner Systems

Abstract

Apartial Steiner (n,r,l)-systemis anr-uniform hypergraph onnvertices in which every set oflvertices is contained in at most one edge. A partial Steiner (n,r,l)-system iscompleteif every set oflvertices is contained in exactly one edge. In a hypergraph, the independence number α() denotes the maximum size of a set of vertices incontaining no edge. In this article we prove the following. Given integersr,lsuch thatr≥ 2l− 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-systemsuch that$$\alpha(\HH) \lesssim \biggl(\frac{l-1}{r-1}(r)_l\biggr)^{\frac{1}{r-1}}n^{\frac{r-l}{r-1}} (\log n)^{\frac{1}{r-1}} \quad \mbox{ as }n \rightarrow \infty.$$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 randomr-uniform hypergraph of the same density. Ifl= 2 orl= 3, then for infinitely manyrthe partial Steiner systems constructed are complete for infinitely manyn.