Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-98-14

June 1998

Balanced Pairs in Partial Orders

Graham R. Brightwell


An  alpha-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  alpha  and   1 - alpha.  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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