On a combinatorial conjecture concerning disjoint occurrences of events
type: Item
instance of: scholarly article
title: On a combinatorial conjecture concerning disjoint occurrences of events
zbMATH Open document ID: 0617.60011
author: Yanyan Li
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Recently [J. Appl. Probab. 22, 556--569 (1985; Zbl 0571.60019)] the first author and \textit{H. Kesten} stated the following conjecture on Bernoulli sequences: Let \(\Omega =\{0,1\}^ n\) (\(n\) a positive integer). For \(K\subset \{1,2,...,n\}\), \(\omega =(\omega_ 1,...,\omega_ n)\in \Omega\) let \([\omega]_ K=\{\omega '\in \Omega |\omega_ i'=\omega_ i\) for all \(i\in K\}\). Finally for \(A,B\subset \Omega\) define \[ A\diamond B=\{\omega \in \Omega | \text{ there exists } K\subset \{1,...,n\},\;[\omega]_ K\subset A,\;[\omega]_{\bar K}\subset B\}, \] the set of ``disjoint realizations'' of \(A\) and \(B\). Is it true that \(2^ n| A\diamond B| \leq | A| \cdot | B|\) (where \(| \cdot |\) denotes cardinality)? In this paper the above conjecture is proved in four nontrivial special cases, some of them are extensions of \textit{T. E. Harris}' FKG inequality [Math. Proc. Camb. Philos. Soc. 56, 13--20 (1960; Zbl 0122.36403)].
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4001078
zbMATH Keywords: correlation inequality
zbMATH Keywords: FKG inequality
zbMATH Keywords: combinatorial probability percolation
zbMATH Keywords: Bernoulli sequences
Wikidata QID: Q123023848
reviewed by: Tamás F. Móri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992274
OpenAlex ID: W2092293898
DOI: 10.1214/AOP/1176992274
description: scientific article; zbMATH DE number 4001078