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On the Independence Number of Steiner Systems

Published online by Cambridge University Press:  30 January 2013

ALEX EUSTIS  and
JACQUES VERSTRAËTE
ALEX EUSTIS
Affiliation:
Department of Mathematics, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA (e-mail: jbaverstraete@gmail.com)
JACQUES VERSTRAËTE
Affiliation:
Department of Mathematics, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA (e-mail: jbaverstraete@gmail.com)
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Abstract

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 , the independence number α() denotes the maximum size of a set of vertices in containing no edge. In this article we prove the following. Given integers r,l such that r ≥ 2l − 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-system such 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 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.

Keywords

Primary 05B05Secondary 05B0705B2005B2505D40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 241 - 252
Copyright
Copyright © Cambridge University Press 2013

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