The 1/3-2/3 Conjecture for $N$-Free Ordered Sets

Imed Zaguia

Abstract


A balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We prove that every finite $N$-free ordered set which is not totally ordered has a balanced pair.

Keywords


Ordered set; Linear extension; $N$-free; Balanced pair; 1/3-2/3 Conjecture

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