The 1/3–2/3 conjecture for linear extensions of posets

Let $P$ be a finite poset that is not a chain. For distinct $x,y\in P$, write $\delta_{xy}=\min\\{p(x\prec y),p(y\prec x)\\}$ and $\delta(P)=\max_{x\ne y}\delta_{xy}$. The 1/3–2/3 conjecture.

$$\delta(P)\geq \frac13.$$

This is a central balancing conjecture for uniformly random linear extensions. It was proposed by Kislitsyn and independently by Fredman and Linial; the conjecture remains open in general, although many special classes of posets are known and a positive universal constant is known.