The 1/3-2/3 conjecture for $N$-free ordered sets

TL;DR

This paper proves that every finite N-free ordered set that is not totally ordered contains a balanced pair, advancing understanding of the 1/3-2/3 conjecture in ordered set theory.

Contribution

It establishes the existence of a balanced pair in all finite N-free ordered sets that are not totally ordered, confirming a special case of the 1/3-2/3 conjecture.

Findings

Every finite N-free ordered set not totally ordered has a balanced pair.

The result applies to a broad class of ordered sets, supporting the 1/3-2/3 conjecture.

The proof advances the theory of linear extensions in ordered sets.

Abstract

A balanced pair in a finite 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 prove that every finite $N$-free ordered set which is not totally ordered has a balanced pair.