{"paper":{"title":"Combined tilings and separated set-systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander V. Karzanov, Gleb A. Koshevoy, Vladimir I. Danilov","submitted_at":"2014-01-24T18:32:06Z","abstract_excerpt":"In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered $n$-element set $[n]$ (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains $D\\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the hyper-simplex $\\{X\\subseteq[n]\\colon |X|=m\\}$ for $m$ fixed), where $D$ is called pure if all maximal weakly separated collections in $D$ have the same cardinality. These conjectures have been answered affirmatively.\n  In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6418","kind":"arxiv","version":3},"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"}