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The $\frac 1 3$–$\frac 2 3$ conjecture

Does every finite partially ordered set that is not totally ordered contain two elements xx and yy such that the probability that xx appears before yy in a random linear extension is between 13\frac 1 3 and 23\frac 2 3?

The set of all total order extensions is represented as order preserving bijections PP of 1,...,n1, ..., n.

Source checked Jul 26, 20261 pinned Lean statementInspect problem