A Family of Partially Ordered Sets with Small Balance Constant

TL;DR

This paper constructs a sequence of finite partially ordered sets with balance constants approaching approximately 0.349, providing new examples that challenge existing bounds related to the 1/3-2/3 conjecture.

Contribution

It introduces a new family of posets with smaller balance constants than previously known, advancing understanding of the 1/3-2/3 conjecture.

Findings

Sequence of posets with balance constants near 0.349

First known examples with balance constants below 1/3

Progress towards understanding the minimal possible balance constant

Abstract

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by \[ \delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$. In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.