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Order Theory
Open
1/3–2/3 Conjecture
In every finite partially ordered set that is not a total order, does there exist an incomparable pair (x, y) such that the fraction of linear extensions with x before y lies between 1/3 and 2/3?
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- Validated increment
Weaker balance constants known; 1/3–2/3 still open
It is known that some incomparable pair is somewhat balanced (constants weaker than 1/3). Pushing the constant all the way to 1/3 is the remaining core.
Previous best
Some pair with balance in (δ, 1−δ) for δ < 1/3
New certified
Best published δ still strictly below 1/3
Public literature baseline
Also open