{"paper":{"title":"A Sparse colorful polytopal KKM Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel McGinnis, Shira Zerbib","submitted_at":"2021-12-29T06:57:56Z","abstract_excerpt":"Recently Sober\\'on proved a far-reaching generalization of the colorful KKM Theorem due to Gale: let $n\\geq k$, and assume that a family of closed sets $(A^i_j\\mid i\\in [n], j\\in [k])$ has the property that for every $I\\in \\binom{[n]}{n-k+1}$, the family $\\big(\\bigcup_{i\\in I}A^i_1,\\dots,\\bigcup_{i\\in I}A^i_k\\big)$ is a KKM cover of the $(k-1)$-dimensional simplex $\\Delta^{k-1}$; then there is an injection $\\pi:[k] \\rightarrow [n]$ so that $\\bigcap_{i=1}^k A_i^{\\pi(i)}\\neq \\emptyset$. We prove a polytopal generalization of this result, answering a question of Sober\\'on in the same note. We als"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14421","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.14421/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}