, Volume 12, Issue 4, pp 327–349

Balancing pairs and the cross product conjecture

  • G. R. Brightwell
  • S. Felsner
  • W. T. Trotter

DOI: 10.1007/BF01110378

Cite this article as:
Brightwell, G.R., Felsner, S. & Trotter, W.T. Order (1995) 12: 327. doi:10.1007/BF01110378


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 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 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.

Mathematics Subject Classifications (1991)


Key words

Partially ordered setlinear extensionbalancing pairscross-product conjectureAhlswede-Daykin inequalitysorting

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • G. R. Brightwell
    • 1
  • S. Felsner
    • 2
  • W. T. Trotter
    • 3
  1. 1.Department of MathematicsLondon School of EconomicsLondonU.K.
  2. 2.Fachbereich Mathematik, Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Department of MathematicsArizona State UniversityTempeU.S.A.