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On proportional transitivity of ordered sets. (English) Zbl 0912.06005
Summary: For a finite partially ordered set (poset) \(P\), denote by \(p (x<y)\) the fraction of linear extensions of \(P\) in which \(x\) precedes \(y\). It is shown that if \(p (x<y)\) and \(p (y< z)\) are each at least \(u\) with \(u\geq \rho\approx 0.78\), then \(p (x<z)\) is at least \(u\). The result stated is mainly a consequence of the \(XYZ\) inequality [L. A. Shepp, Ann. Probab. 10, 824-827 (1982; Zbl 0484.60010)] and a theorem of K. Ball [Stud. Math. 88, 69-84 (1988; Zbl 0642.52011)] which allows us to reduce to a two-dimensional version of the problem.

MSC:
06A07 Combinatorics of partially ordered sets
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