Balancing Pairs and the Cross Product Conjecture

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 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 Classification (1991). 06A07, 06A10.