, Volume 12, Issue 4, pp 327–349 | Cite as

Balancing pairs and the cross product conjecture

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


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)

06A07 06A10 

Key words

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


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  1. 1.
    Ahlswede, R. and Daykin, D. E. (1978) An inequality for the weights of two families of sets, their unions and intersections,Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43, 183–185.Google Scholar
  2. 2.
    Brightwell, G. (1988) Linear extensions of infinite posets,Discrete Mathematics 70, 113–136.Google Scholar
  3. 3.
    Brightwell, G. (1989) Semiorders and the 1/3–2/3 conjecture,Order 5, 369–380.Google Scholar
  4. 4.
    Brightwell, G. (1990) Events correlated with respect to every subposet of a fixed poset,Graphs and Combinatorics 6, 111–131.Google Scholar
  5. 5.
    Brightwell, G. and Wright, C. D. (1992) The 1/3–2/3 conjecture for 5-thin posets,SIAM J. Discrete Math. 5, 467–474.Google Scholar
  6. 6.
    Felsner, S. and Trotter, W. T. (1993) Balancing pairs in partially ordered sets, inCombinatorics, Vol. 1, Paul Erdos is Eighty, pp. 145–157.Google Scholar
  7. 7.
    Fishburn, P. C. (1984) A correlational inequality for linear extensions of a poset,Order 1, 127–137.Google Scholar
  8. 8.
    Fishburn, P., Gehrlein, W. G., and Trotter, W. T. (1992) Balance theorems for height-2 posets,Order 9, 43–53.Google Scholar
  9. 9.
    Fredman, M. (1976) How good is the information theoretic bound in sorting?Theoretical Computer Science 1, 355–361.Google Scholar
  10. 10.
    Friedman, J. (1993) A Note on Poset Geometries,SIAM J. Computing 22, 72–78.Google Scholar
  11. 11.
    Kahn, J. and Kim, J. Entropy and sorting,JACM, to appear.Google Scholar
  12. 12.
    Kahn, J. and Linial, N. (1991) Balancing extensions via Brunn-Minkowski,Combinatorica 11, 363–368.Google Scholar
  13. 13.
    Kahn, J. and Saks, M. (1984) Balancing poset extensions,Order 1, 113–126.Google Scholar
  14. 14.
    Khachiyan, L. (1989) Optimal algorithms in convex programming decomposition and sorting, in J. Jaravlev (ed.),Computers and Decision Problems, Nauka, Moscow, pp. 161–205 (in Russian).Google Scholar
  15. 15.
    Kislitsyn, S.S. (1968) Finite partially ordered sets and their associated sets of permutations,Matematicheskiye Zametki 4, 511–518.Google Scholar
  16. 16.
    Komlós, J. (1990) A strange pigeon-hole principle,Order 7, 107–113.Google Scholar
  17. 17.
    Linial, N. (1984) The information theoretic bound is good for merging,SIAM J. Computing 13, 795–801.Google Scholar
  18. 18.
    Saks, M. (1985) Balancing linear extensions of ordered sets,Order 2, 327–330.Google Scholar
  19. 19.
    Shepp, L. A. (1980) The FKG inequality and some monotonicity properties of partial orders,SIAM J. Alg. Disc. Meths. 1, 295–299.Google Scholar
  20. 20.
    Trotter, W. T. (1995) Partially ordered sets, in R. L. Graham, M. Grötschel, and L. Lovász (eds),Handbook of Combinatorics, to appear.Google Scholar
  21. 21.
    Trotter, W. T. (1991)Combinatorics and Partially Ordered Sets: Dimension Theory, The Johns Hopkins University Press, Baltimore, MD.Google Scholar

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.

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