The 1/3-2/3 Conjecture for ordered sets whose cover graph is a forest

Abstract

A balanced pair in an ordered set $P=(V,\le )$ 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.