## Abstract

In a finite partially ordered set, Prob (*x*>*y*) denotes the proportion of linear extensions in which element*x* appears above element*y*. 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 3/11<Prob(*x*>*y*)<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 paper, we show that there exists a pair (*x,y*) for which (5−√5)/10⩽Prob(*x*>*y*)⩽(5+√5)/10. The proof depends on an application of the Ahlswede-Daykin inequality to prove a special case of a conjecture which we call the Cross Product Conjecture. Our proof also requires the full force of the Kahn-Saks approach — in particular, it requires the Alexandrov-Fenchel inequalities for mixed volumes.

We extend our result on balancing pairs to a class of countably infinite partially ordered sets where the 1/3–2/3 conjecture is*false*, and our bound is best possible. Finally, we obtain improved bounds for the time required to sort using comparisons in the presence of partial information.

## Mathematics Subject Classifications (1991)

06A07 06A10## Key words

Partially ordered set linear extension balancing pairs cross-product conjecture Ahlswede-Daykin inequality sorting## Preview

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