The minimum independence number for designs
type: Item
instance of: scholarly article
title: The minimum independence number for designs
zbMATH Open document ID: 0824.05005
DOI: 10.1007/BF01200754
author: David A. Grable
author: Kevin T. Phelps
author: Vojtěch Rödl
published in: Combinatorica
publication date: 1995-07-24T00:00:00Z
review text: Let \(V\) be a set with \(n\) elements, and let \(D\) denote a collection of subsets of \(V\), each subset with \(k\) elements in it, such that every subset of \(t\) elements of \(V\) appears exactly in \(\lambda\) blocks of \(D\). Such a configuration is called \(t\)-\((n, k, \lambda)\) design and is denoted by a pair \(P= (V, D)\). If no block of \(D\) is contained in a subset \(S\) of \(V\), then \(S\) is called independent in \(P\). The maximum cardinality of any independent set in \(P\) is called the independence number and is denoted by \(\alpha(P)\). This paper deals with the problem of obtaining an upper bound on the minimum independence number of all designs of order \(n\). The main result is stated as follows: Consider designs with \(t= 2, 3\) and let \(k\geq 2t- 1\). There exists a constant \(c\) such that if \(q\) is a sufficiently large prime power and if there exists a \(t\)-\((q+ t- 2, k, \lambda)\) design then there exists a \(t\)-\((n, k, \lambda)\) design \(P\) with \(n= q^ 2+ t- 2\) such that \(\alpha(P)\leq c(n)^{{k- t\over k- 1}}(\ln n)^{{1\over k- 1}}\), where \(c\) is not greater than a certain number \(N(k, t, \lambda)\).
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 778332
zbMATH Keywords: block
zbMATH Keywords: independent set
zbMATH Keywords: independence number
zbMATH Keywords: minimum independence number
zbMATH Keywords: designs
reviewed by: D. V. Chopra
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01200754
OpenAlex ID: W2027121235
description: scientific article; zbMATH DE number 778332