Families implying the Frankl conjecture
type: Item
instance of: scholarly article
title: Families implying the Frankl conjecture
zbMATH Open document ID: 1021.05096
author: Theresa P. Vaughan
published in: European Journal of Combinatorics
publication date: 2003-03-18T00:00:00Z
review text: A union-closed (UC) family \({\mathcal A}\) is a finite collection of sets such that the union of any two members of \({\mathcal A}\) is also a member of \({\mathcal A}\). It was conjectured by \textit{P. Frankl} [Discrete Math. 26, 111-118 (1979; Zbl 0397.05004)] that, for any such \({\mathcal A}\), there must be an element \(x\) in at least half of the sets of \({\mathcal A}\). This conjecture is still open. Define a \(\text{FC}(n)\)-family to be a collection \({\mathcal B}\) of sets, \(|S|= n\) where \(S= \bigcup_{B\in{\mathcal B}} B\), such that, if \({\mathcal A}\) is any UC-family containing \({\mathcal B}\), then one of the elements of \(S\) is contained in at least half of the members of \({\mathcal A}\). Such a \({\mathcal B}\) which does not contain any \(\text{FC}(m)\)-family for any \(m< n\) is called a proper \(\text{FC}(n)\)-family. The author obtains proper \(\text{FC}(n)\)-families for \(n= 5,6\) and \(7\), making use of results of \textit{B. Poonen} [J. Comb. Theory, Ser. A 59, 253-268 (1992; Zbl 0758.05096)].
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 1884171
zbMATH Keywords: union-closed family
MaRDI profile type: Publication
full work available at URL: eujc.2002.0586
OpenAlex ID: W2047963661
Wikidata QID: Q123111305
cites work: Q3741626
cites work: Q4320677
cites work: On union-closed families. I
cites work: Union-closed families
cites work: Q4204156
DOI: 10.1006/EUJC.2002.0586
description: scientific article; zbMATH DE number 1884171