A family of inequalities for intersecting antichains of subsets of an \(n\)-set
type: Item
instance of: scholarly article
title: A family of inequalities for intersecting antichains of subsets of an \(n\)-set
zbMATH Open document ID: 0842.05093
published in: European Journal of Combinatorics
publication date: 1996-03-18T00:00:00Z
review text: Let \(F= \{(A_1, B_1),\dots, (A_m, B_m)\}\) be a family of pairs of subsets of \(N= \{1,\dots, n\}\). This family is called pair-intersecting if for all \(i, j\in \{1,\dots, m\}\), \(A_i\cap A_j\neq \varnothing\) as well as \(A_i\cap B_j= \varnothing\) iff \(i= j\). The authors prove a theorem of the following kind: Let \(f\) be a non-negative real-valued function defined on the power set of \(N\) satisfying several technical conditions and let \(F\) be a pair-intersecting family. Then \[ \sum^m_{i= 1} {f(A_i)\over (\begin{smallmatrix} |A_i|+ |B_i|\\ |A_i|\end{smallmatrix}) (\begin{smallmatrix} n\\ |A_i|+ |B_i|\end{smallmatrix})}\leq 1. \] The proof is based on the cyclic permutation method. The authors also derive several inequalities for intersecting antichains in the Boolean lattice, but it should be mentioned that ``all'' (i.e. facet defining) inequalities are known by a result of \textit{G. O. H. Katona} and \textit{G. Schild} [Linear inequalities describing the class of intersecting Sperner families of subsets. I. Topics in combinatorics and graph theory. Essays in honour of Gerhard Ringel, 413-420 (1990; Zbl 0748.05005)].
reviewed by: Konrad Engel
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 05A20
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 847632
zbMATH Keywords: LYM-inequality
zbMATH Keywords: pair-intersecting family
zbMATH Keywords: cyclic permutation
zbMATH Keywords: intersecting antichains
zbMATH Keywords: Boolean lattice
zbMATH Keywords: intersecting Sperner families
author: Changyu Wang
author: Huishan Zhou
MaRDI profile type: Publication
full work available at URL: eujc.1996.0007
OpenAlex ID: W2085375236
DOI: 10.1006/EUJC.1996.0007
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description: scientific article; zbMATH DE number 847632