Frankl's conjecture and the dual covering property
type: Item
instance of: scholarly article
title: Frankl's conjecture and the dual covering property
zbMATH Open document ID: 1202.06001
author: R. S. Shewale
author: Vinayak Joshi
author: V. S. Kharat
published in: Graphs and Combinatorics
publication date: 2009-12-09T00:00:00Z
review text: Frankl's conjecture states that for any union-closed family of subsets \(F\) of a finite set \(X\) with at least two members there exists an element \(x\) of \(X\) such that at least half of the members of \(F\) contain \(x\). B. Poonen reformulated this for lattices as follows: Let \(L\) be a finite lattice with at least two elements. Then there is a join-irreducible element \(j\) such that the number of elements in the principal filter generated by \(j\) does not exceed half of the size of \(L\). The present authors generalize this for posets where they define that an element \(j\) of a poset is join-irreducible if \(j\) covers a unique element. It was already proved that Frankl's conjecture for lattices is valid for some large classes of lattices including modular lattices, relatively complemented lattices or lower semimodular lattices. The main result here is: Frankl's conjecture for posets is true for posets satisfying the dual covering property. Such posets are treated in the paper.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 5644623
zbMATH Keywords: dual covering property
zbMATH Keywords: Frankl's Conjecture
reviewed by: Ivan Chajda
MaRDI profile type: Publication
full work available at URL: s00373-008-0820-x
OpenAlex ID: W2125677154
Wikidata QID: Q123219998
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DOI: 10.1007/S00373-008-0820-X
description: scientific article; zbMATH DE number 5644623