The 1/3–2/3 Conjecture for finite posets

Let $P$ be a finite poset on $n$ elements. A linear extension is an order-preserving bijection $bb:P\to[n]$, and for $x,y\in P$ define

$$\delta_P(x,y)=\frac{|\{\text{linear extensions }\lambda:P\to[n]\text{ such that }\lambda(x)>\lambda(y)\}|}{|\{\text{linear extensions of }P\}|}.$$

The balance constant of $P$ is

$$b(P)=\max_{x,y\in P}\min(\delta_P(x,y),1-\delta_P(x,y)).$$

The 1/3–2/3 Conjecture. For any finite poset $P$ which is not a total order, $b(P)\geq\frac{1}{3}$.

This is a longstanding open problem in the theory of posets, with the bound proved for several special classes, including width-two posets. The conjecture asks for a universal information-theoretic lower bound on how balanced some pair of incomparable elements must be.