{"paper":{"title":"Weak Separation, Pure Domains and Cluster Distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Miriam Farber, Pavel Galashin","submitted_at":"2016-12-16T07:17:45Z","abstract_excerpt":"Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a variety of wider examples of purity in different settings. In this paper we consider the collection $\\mathcal A_{I,J}$ of sets that are weakly separated from two fixed sets $I$ and $J$. We show that all maximal by inclusion weakly separated collections $\\mathcal W\\subset\\mathcal A_{I,J}$ are also maximal by size, provided that $I$ and $J$ are sufficiently \"generic\". We also give a simple formula for the cardinality of $\\mathcal W$ in terms of $I$ and $J$. We apply our result to calculate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05387","kind":"arxiv","version":1},"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"}