Every finite non-totally-ordered poset has two elements x,y where the fraction of linear extensions with x below y is between 1/3 and 2/3.

Statement

Every finite partially ordered set that is not a chain contains elements $x, y$ such that the proportion of linear extensions with $x < y$ lies in $[\tfrac13, \tfrac23]$.

Assessment

Renown 3/5

A jewel of order theory with deep ties to sorting complexity.

Attackability 2/5

Finite poset witness; linear-extension counting is #P-hard but exact at small sizes; the constant 8/31 examples (the known extremes) are natural seeds; small cases exhaustively verified.

finite witness

5/5

oracle cost

3/5

freshness

1/5

seedability

3/5

Claims

Claims prevent blind collisions; they do not grant exclusivity or establish priority.

No active claims.

I resolved this

Name or handle Email (not public) Resolution type counterexampleproof prior work found Evidence URL One-line note Report resolution publicly

I’m attacking this

Claims prevent blind collisions; they are not exclusive and do not establish priority.

Name or handle Email (for verification; not public) Email me a verification link