Linear dependencies among subsets of a finite set
type: Item
instance of: scholarly article
title: Linear dependencies among subsets of a finite set
zbMATH Open document ID: 0532.05006
DOI: 10.1016/S0195-6698(83)80027-4
published in: European Journal of Combinatorics
publication date: 1983-01-01T00:00:00Z
review text: Let integers \(k\) and \(t\) be given so that \(k>t\ge 0\). A family of \(k\)-sets \(F_1,\ldots,F_m\) with rational coefficients \(\alpha_1,\ldots,\alpha_m\) forms a \(0\)-design for \(k\) and \(t\) if, for any \(t\)-set \(G\), \(\sum \alpha_i=0\) (where the sum is taken over all \(i\) such that \(G\subset F_i)\). In this paper, the authors prove that, if \(\binom{t+1}{1}, \ldots, \binom{t+1}{k-t-1}\) have a common divisor \(d\) which does not divide \(\alpha_i\), then there exists an index \(j\) so that \(| F_i\cap F_j| = k-t-1\). Using this result, the authors deduce some other results in extremal set theory relating to problems posed by Erdős.
reviewed by: Jack E. Graver
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 3843755
zbMATH Keywords: family of sets
zbMATH Keywords: 0-design
zbMATH Keywords: extremal set theory
Wikidata QID: Q114656088
author: Peter Frankl
author: N. M. Singhi
MaRDI profile type: Publication
cites work: Q4198534
cites work: Q4134005
cites work: INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS
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cites work: On the Structure oft-Designs
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cites work: Intersection theorems for systems of finite sets
description: scientific article; zbMATH DE number 3843755