On a combinatorial conjecture concerning disjoint occurrences of events
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)].
scientific article; zbMATH DE number 4001078