{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:7OQZBLLXTOHU52XKMQKOCLP23Z","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0add5234645d5da78612762a83c921122943cafa4955d5da74f7e47746c0ac98","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-12-29T06:57:56Z","title_canon_sha256":"efdb91b1b834f83ad38ca71f75fe4c02800d40283c6363974073149bbb50d9ec"},"schema_version":"1.0","source":{"id":"2112.14421","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.14421","created_at":"2026-07-05T03:44:22Z"},{"alias_kind":"arxiv_version","alias_value":"2112.14421v1","created_at":"2026-07-05T03:44:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.14421","created_at":"2026-07-05T03:44:22Z"},{"alias_kind":"pith_short_12","alias_value":"7OQZBLLXTOHU","created_at":"2026-07-05T03:44:22Z"},{"alias_kind":"pith_short_16","alias_value":"7OQZBLLXTOHU52XK","created_at":"2026-07-05T03:44:22Z"},{"alias_kind":"pith_short_8","alias_value":"7OQZBLLX","created_at":"2026-07-05T03:44:22Z"}],"graph_snapshots":[{"event_id":"sha256:2a0eaf7c4e7de4a0d3dfe10a6b05fee4e6c3a1b714f9d33ceb19d584e2907ad0","target":"graph","created_at":"2026-07-05T03:44:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2112.14421/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Daniel McGinnis, Shira Zerbib","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-12-29T06:57:56Z","title":"A Sparse colorful polytopal KKM Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14421","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:178f58a9d3f48b848f180a73c6e9cc74f70de8f78e60cab61c56bfa91508d341","target":"record","created_at":"2026-07-05T03:44:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0add5234645d5da78612762a83c921122943cafa4955d5da74f7e47746c0ac98","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-12-29T06:57:56Z","title_canon_sha256":"efdb91b1b834f83ad38ca71f75fe4c02800d40283c6363974073149bbb50d9ec"},"schema_version":"1.0","source":{"id":"2112.14421","kind":"arxiv","version":1}},"canonical_sha256":"fba190ad779b8f4eeaea6414e12dfade6b9813aa5877126e64cfbea8efe9a360","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fba190ad779b8f4eeaea6414e12dfade6b9813aa5877126e64cfbea8efe9a360","first_computed_at":"2026-07-05T03:44:22.028199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:44:22.028199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bCpb17ICS+S+AMLGM4DE+IOZ5a/GqUdysbfePSiTFx0dRjstgw9ec5rj5xkyYA6Bdqsog459Qi8h8leQI/afDA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:44:22.028564Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.14421","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:178f58a9d3f48b848f180a73c6e9cc74f70de8f78e60cab61c56bfa91508d341","sha256:2a0eaf7c4e7de4a0d3dfe10a6b05fee4e6c3a1b714f9d33ceb19d584e2907ad0"],"state_sha256":"c41f9552c6c1687859f826a2eeffbdc9ac6c72a6ccabe592ced56d1a16466b75"}