# Balancing pairs and the cross product conjecture

@article{Brightwell1995BalancingPA, title={Balancing pairs and the cross product conjecture}, author={Graham R. Brightwell and Stefan Felsner and William T. Trotter}, journal={Order}, year={1995}, volume={12}, pages={327-349} }

In a finite partially ordered set, Prob (x>y) denotes the proportion of linear extensions in which elementx appears above elementy. In 1969, S. S. Kislitsyn conjectured that in every finite poset which is not a chain, there exists a pair (x,y) for which 1/3⩽Prob(x>y)⩽2/3. In 1984, J. Kahn and M. Saks showed that there exists a pair (x,y) with 3/11y)<8/11, but the full 1/3–2/3 conjecture remains open and has been listed among ORDER's featured unsolved problems for more than 10 years.In this…

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