Balanced pairs in partial orders Original Research Article

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 13–23 Conjecture states that, in every finite partial order P, not a chain, there is a 13-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.