{"paper":{"title":"On the $1/3-2/3$ Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bruce E. 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. We make progress on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.04985","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}