Greedy balanced pairs in N-free ordered sets

An $\alpha$-greedy balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of greedy linear extensions of $P$ that put $x$ before $y$ among all greedy linear extensions is in the real interval $[\alpha, \alpha-1]$. We prove that every $N$-free ordered set which is not totally ordered has a $\frac{1}{2}$-greedy balanced pair.

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