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A balanced pair in an ordered set P = (V, ≤) is a pair (x, y) of elements of V such that the proportion of linear extensions of P that put x before y is in the real interval [1/3, 2/3]. We define the notion of a good pair and claim any ordered set that has a good pair will satisfy the conjecture and furthermore every ordered set which is not totally ordered and has a forest as its cover graph has a good pair.

The 1/3-2/3 Conjecture for Ordered Sets whose Cover Graph is a Forest