{"paper":{"title":"A bound on partitioning clusters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Kane, Terence Tao","submitted_at":"2017-02-03T05:48:28Z","abstract_excerpt":"Let $X$ be a finite collection of sets (or \"clusters\"). We consider the problem of counting the number of ways a cluster $A \\in X$ can be partitioned into two disjoint clusters $A_1, A_2 \\in X$, thus $A = A_1 \\uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound $$ | \\{ (A_1,A_2,A) \\in X \\times X \\times X: A = A_1 \\uplus A_2 \\} | \\leq |X|^{3/p} $$ where $|X|$ denotes the cardinality of $X$, and $p := \\log_3 \\frac{27}{4} = 1.73814\\dots$, so that $\\frac{3}{p} = 1.72598\\dots$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.00912","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"}