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

TL;DR

This paper proves the 1/3-2/3 Conjecture for ordered sets with a forest cover graph by introducing the concept of good pairs, showing such sets always contain one, thus satisfying the conjecture.

Contribution

It introduces the notion of good pairs and proves that all ordered sets with a forest cover graph have such pairs, confirming the conjecture in this case.

Findings

Ordered sets with forest cover graphs have good pairs.

Such sets satisfy the 1/3-2/3 Conjecture.

The concept of good pairs aids in proving the conjecture.

Abstract

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