Abstract
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.
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