Balanced pairs in partial orders
Brightwell, G.
(1999).
Balanced pairs in partial orders.
Discrete Mathematics,
201(1-3), 25-52.
https://doi.org/10.1016/S0012-365X(98)00311-2
(1999).
Balanced pairs in partial orders.
Discrete Mathematics,
201(1-3), 25-52.
https://doi.org/10.1016/S0012-365X(98)00311-2
An α-balanced pair in a partially ordered set P = (X, <) is a pair (x, y) of elements of X such that the proportion of linear extensions of P with x below y lies between α and 1 − α. The 1/3–2/3 Conjecture states that, in every finite partial order P, not a chain, there is a 1/3-balanced pair. This was first conjectured in a 1968 paper of Kislitsyn, and remains unsolved. We survey progress towards a resolution of the conjecture, and discuss some of the many related problems.