Abstract
A well-known conjecture of Fredman is that, for every finite partially ordered set (X, <) which is not a chain, there is a pair of elements x, y such that P(x<y), the proportion of linear extensions of (X, <) with x below y, lies between 1/3 and 2/3. In this paper, we prove the conjecture in the special case when (X, <) is a semiorder. A property we call 2-separation appears to be crucial, and we classify all locally finite 2-separated posets of bounded width.
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Communicated by W. T. Trotter
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Brightwell, G.R. Semiorders and the 1/3–2/3 conjecture. Order 5, 369–380 (1989). https://doi.org/10.1007/BF00353656
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DOI: https://doi.org/10.1007/BF00353656