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For $0\\leq \\alpha \\leq \\frac{1}{2}$, we say $(x,y)$ is an $\\alpha$-balanced pair if $\\alpha \\leq \\mathbb{P}(x\\prec y) \\leq 1-\\alpha.$ The $1/3-2/3$ Conjecture states that every finite partially ordered set which is not a chain has a $1/3$-balanced pair. 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Sagan, Emily J. Olson","submitted_at":"2017-06-15T17:38:48Z","abstract_excerpt":"Let $(P,\\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\\cdots x_n$ in one-line notation. For distinct elements $x,y\\in P$, we define $\\mathbb{P}(x\\prec y)$ to be the proportion of linear extensions of $P$ in which $x$ comes before $y$. For $0\\leq \\alpha \\leq \\frac{1}{2}$, we say $(x,y)$ is an $\\alpha$-balanced pair if $\\alpha \\leq \\mathbb{P}(x\\prec y) \\leq 1-\\alpha.$ The $1/3-2/3$ Conjecture states that every finite partially ordered set which is not a chain has a $1/3$-balanced pair. 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