{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BADD4SR7EBVLWCNDHDZEXAHA2I","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":"0f3f1508bbd9a5d6804969670696446f77468e4390dfc7fa063d491e1ffade50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-30T20:27:21Z","title_canon_sha256":"d688fc542741c33bc038fc97fccca88e6a164c16852914f43222a5562bdac605"},"schema_version":"1.0","source":{"id":"1307.8138","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.8138","created_at":"2026-05-18T03:17:02Z"},{"alias_kind":"arxiv_version","alias_value":"1307.8138v1","created_at":"2026-05-18T03:17:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.8138","created_at":"2026-05-18T03:17:02Z"},{"alias_kind":"pith_short_12","alias_value":"BADD4SR7EBVL","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"BADD4SR7EBVLWCND","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"BADD4SR7","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:77fc52460e3220cdcd992dd2e73a3b6a86b20fe45b1eaf4bd72ad1327b586839","target":"graph","created_at":"2026-05-18T03:17:02Z","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"},"paper":{"abstract_excerpt":"Intuitively, a tangle of large order in a graph is a highly-connected part of the graph, and it is known that if a graph has a tangle of large order then it has a large grid minor. Here we show that for any k, if G has a tangle of large order and Z is a set of vertices of cardinality k that cannot be separated from the tangle by any separation of order less than k, then G has a large grid minor containing Z, in which the members of Z all belong to the outside of the grid. This is a lemma for use in a later paper.","authors_text":"D\\'aniel Marx, Paul Seymour, Paul Wollan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-30T20:27:21Z","title":"Rooted grid minors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.8138","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":"sha256:0369fbfc6059fbeb79419c2cf170186adad4be612811941e11f24a882eaf7c42","target":"record","created_at":"2026-05-18T03:17:02Z","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":"0f3f1508bbd9a5d6804969670696446f77468e4390dfc7fa063d491e1ffade50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-30T20:27:21Z","title_canon_sha256":"d688fc542741c33bc038fc97fccca88e6a164c16852914f43222a5562bdac605"},"schema_version":"1.0","source":{"id":"1307.8138","kind":"arxiv","version":1}},"canonical_sha256":"08063e4a3f206abb09a338f24b80e0d22f01999192968b169bb6be09d7d64d30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08063e4a3f206abb09a338f24b80e0d22f01999192968b169bb6be09d7d64d30","first_computed_at":"2026-05-18T03:17:02.291574Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:02.291574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8STlMAfLNxtpkFEp+nU8Z1TcEqzaFILKckHyr+zI28Yh9JNRE7MoF9t5FnYYnVx9qbrSsMdex1e/yECH4JWLDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:02.292242Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.8138","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0369fbfc6059fbeb79419c2cf170186adad4be612811941e11f24a882eaf7c42","sha256:77fc52460e3220cdcd992dd2e73a3b6a86b20fe45b1eaf4bd72ad1327b586839"],"state_sha256":"ae018ba0bb7c5d6a0a3a699ad5f8eb0e7f17efc833eaf4e7efec5f52000c80de"}