A Family of Partially Ordered Sets with Small Balance Constant

Given a finite poset $\mathcal{P}$ and two distinct elements $x$ and $y$, we let ${\mathrm{pr}}_{\mathcal{P}}(x\prec y)$ 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 $$\delta (\mathcal{P})=\underset{x\mathrm{\ne }y\in \mathcal{P}}{\max }\min \{{\mathrm{pr}}_{\mathcal{P}}(x\prec y),{\mathrm{pr}}_{\mathcal{P}}(y\prec x)\}\mathrm{.}$$ The $1\mathrm{/}3$-$2\mathrm{/}3$ conjecture asserts that $\delta (\mathcal{P})\ge \frac{1}{3}$ 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 $1\mathrm{/}3$. 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.