An extremal set theoretical characterization of some Steiner systems
type: Item
instance of: scholarly article
title: An extremal set theoretical characterization of some Steiner systems
zbMATH Open document ID: 0535.05016
DOI: 10.1007/BF02579293
published in: Combinatorica
publication date: 1983-01-01T00:00:00Z
review text: For n, k, t integers and \(n>k>t\geq 0\), let m(n,k,t) denote the maximum number of sets in a family of k-subsets of an n-set, no two of which intersect in exactly t elements. \textit{P. Erdős} [Proc. 5th Br. comb. Conf., Aberdeen 1975, 169-192 (1976; Zbl 0335.05002)] first posed the problem of determining m(n,k,t); conjecturing that for \(k>2t+1\) and \(n>n_ 0(k,t)\), \(m(n,k,t)=\left( \begin{matrix} n-t-1\\ k-t-1\end{matrix} \right).\) (A proof of this has been announced by the author and Z. Füredi.) In this paper the author shows that for \(k\leq 2t+1\) and k-t prime, \(m(n,k,t)\leq\left( \begin{matrix} n\\ t\end{matrix} \right)\left( \begin{matrix} 2k-t-1\\ k\end{matrix} \right)/\left( \begin{matrix} 2k-t-1\\ t\end{matrix} \right);\) with equality holding if and only if an (n,2k-t-1,t)-Steiner system exists. A linear algebraic approach is used.
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05C65
zbMATH DE Number: 3849244
zbMATH Keywords: subset
zbMATH Keywords: Steiner system
author: Peter Frankl
reviewed by: Katherine Heinrich
MaRDI profile type: Publication
cites work: INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS
cites work: On families of finite sets no two of which intersect in a singleton
cites work: Extremal problems and coverings of the space
cites work: Intersection theorems with geometric consequences
full work available at URL: https://doi.org/10.1007/bf02579293
OpenAlex ID: W1969308840
description: scientific article; zbMATH DE number 3849244