Balancing pairs and the cross product conjecture
 G. R. Brightwell,
 S. Felsner,
 W. T. Trotter
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Abstract
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 3/11<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 AhlswedeDaykin 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 KahnSaks approach — in particular, it requires the AlexandrovFenchel 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 isfalse, 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.
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 Title
 Balancing pairs and the cross product conjecture
 Journal

Order
Volume 12, Issue 4 , pp 327349
 Cover Date
 19951201
 DOI
 10.1007/BF01110378
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 06A07
 06A10
 Partially ordered set
 linear extension
 balancing pairs
 crossproduct conjecture
 AhlswedeDaykin inequality
 sorting
 Industry Sectors
 Authors

 G. R. Brightwell ^{(1)}
 S. Felsner ^{(2)}
 W. T. Trotter ^{(3)}
 Author Affiliations

 1. Department of Mathematics, London School of Economics, Houghton Street, WC2A 2AE, London, U.K.
 2. Fachbereich Mathematik, Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany
 3. Department of Mathematics, Arizona State University, 85287, Tempe, AZ, U.S.A.