The 1/3-2/3 conjecture for finite posets

Prove that every finite poset that is not totally ordered contains a pair of incomparable elements whose sorting probability lies between 1/3 and 2/3, equivalently, establish that for every non-chain finite poset P, \[\max_{x,y\in P}\min(Pr_P(x,y),Pr_P(y,x))\geq \frac13.\]

Background

The paper studies sorting probabilities of finite posets, defined by the relative numbers of linear extensions placing one element before another. The 1/3-2/3 conjecture asserts that every finite poset that is not a chain has a 1/3-balanced incomparable pair. The conjecture is known for several substantial classes of posets, including posets of height two, width two, and cell posets of partitions and skew partitions, but is unresolved in general.

References

The conjecture remains open, although it has been settled for large classes of posets, such as posets with \le 11 elements, posets where each element is incomparable to at most 6 others, posets with height 2, posets with width two, semiorders, polytrees N-free posets, and posets whose Hasse diagram is a tree.

— Blocking Ideals: a method for filtering linear extensions of a finite poset  (2501.11073 - Jaldevik et al., 19 Jan 2025) in Section 1.2, “Sorting probabilities and the 1/3-2/3 conjecture”

The classical $1/3$--$2/3$ conjecture, first posed by \textcite{Kislitsyn} and later independently by \textcite{Fredman} and \textcite{Linial}, asserts that $\delta(P)\ge 1/3$ whenever $P$ is not a chain.

— On the Gap of Finite Posets  (2608.12678 - Haqi, 13 Aug 2026) in Section 2, Related work

\textcite{KahnSaks} proposed the asymptotic conjecture that $\delta(P)\to1/2$ uniformly as $w(P)\to\infty$.

— On the Gap of Finite Posets  (2608.12678 - Haqi, 13 Aug 2026) in Section 2, Related work