{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:77WIERRQZ6VCGLARE4Y5XDVQ2T","short_pith_number":"pith:77WIERRQ","canonical_record":{"source":{"id":"1101.6006","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-31T16:21:38Z","cross_cats_sorted":["cs.CG","cs.DM","math.AT","math.MG"],"title_canon_sha256":"327af055bc1ba250f3d14156245dd7e36c0540a2e7938922526383a4458b5b0d","abstract_canon_sha256":"0a96cbea9aee00c18e2cabcd7c45f248fc7db0543d6e5036c716aed85f41fa08"},"schema_version":"1.0"},"canonical_sha256":"ffec824630cfaa232c112731db8eb0d4d55a40e1ff29c3a8d1eb2d5d78e0c138","source":{"kind":"arxiv","id":"1101.6006","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.6006","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"arxiv_version","alias_value":"1101.6006v2","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.6006","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"pith_short_12","alias_value":"77WIERRQZ6VC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"77WIERRQZ6VCGLAR","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"77WIERRQ","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:77WIERRQZ6VCGLARE4Y5XDVQ2T","target":"record","payload":{"canonical_record":{"source":{"id":"1101.6006","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-31T16:21:38Z","cross_cats_sorted":["cs.CG","cs.DM","math.AT","math.MG"],"title_canon_sha256":"327af055bc1ba250f3d14156245dd7e36c0540a2e7938922526383a4458b5b0d","abstract_canon_sha256":"0a96cbea9aee00c18e2cabcd7c45f248fc7db0543d6e5036c716aed85f41fa08"},"schema_version":"1.0"},"canonical_sha256":"ffec824630cfaa232c112731db8eb0d4d55a40e1ff29c3a8d1eb2d5d78e0c138","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:57.808428Z","signature_b64":"B7F4AqVl0DZ/7J1xx25xlWpA9T4f7lsyYjB51YiWo5+NJUQ2ynqCFLgJCjfUQ4TG9ue3WQpE7toy94Xiz4KtDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffec824630cfaa232c112731db8eb0d4d55a40e1ff29c3a8d1eb2d5d78e0c138","last_reissued_at":"2026-05-18T04:27:57.807705Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:57.807705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.6006","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M9EO8I4zEStCV5C7k73cHSmCZ/PsSvmrHYnqEQ1gM7etG4Yb7HaHdL0X5cUApo5Ji8kDHNajZKAzxqqaGeDCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T23:06:06.743861Z"},"content_sha256":"31be1da5838148326332667b37937fd1905434e501fde239a0c85152654839e4","schema_version":"1.0","event_id":"sha256:31be1da5838148326332667b37937fd1905434e501fde239a0c85152654839e4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:77WIERRQZ6VCGLARE4Y5XDVQ2T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Helly numbers of acyclic families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM","math.AT","math.MG"],"primary_cat":"math.CO","authors_text":"\\'Eric Colin de Verdi\\`ere, Gr\\'egory Ginot, Xavier Goaoc","submitted_at":"2011-01-31T16:21:38Z","abstract_excerpt":"The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In partic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.6006","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ViQ73mlJXlidLx4t3lOnwqMIXO3W+I4fIkX9D/6rUdqpZ0dqSYZgnNrOAzp3GKGEsK5ChouG8UiqVEh5sXYRDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T23:06:06.744240Z"},"content_sha256":"72f3dfe98cbc5d6d02bad09b0ababaaefaec47304108b1c61e911a9f42ebf372","schema_version":"1.0","event_id":"sha256:72f3dfe98cbc5d6d02bad09b0ababaaefaec47304108b1c61e911a9f42ebf372"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/bundle.json","state_url":"https://pith.science/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T23:06:06Z","links":{"resolver":"https://pith.science/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T","bundle":"https://pith.science/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/bundle.json","state":"https://pith.science/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/77WIERRQZ6VCGLARE4Y5XDVQ2T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:77WIERRQZ6VCGLARE4Y5XDVQ2T","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":"0a96cbea9aee00c18e2cabcd7c45f248fc7db0543d6e5036c716aed85f41fa08","cross_cats_sorted":["cs.CG","cs.DM","math.AT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-31T16:21:38Z","title_canon_sha256":"327af055bc1ba250f3d14156245dd7e36c0540a2e7938922526383a4458b5b0d"},"schema_version":"1.0","source":{"id":"1101.6006","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.6006","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"arxiv_version","alias_value":"1101.6006v2","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.6006","created_at":"2026-05-18T04:27:57Z"},{"alias_kind":"pith_short_12","alias_value":"77WIERRQZ6VC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"77WIERRQZ6VCGLAR","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"77WIERRQ","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:72f3dfe98cbc5d6d02bad09b0ababaaefaec47304108b1c61e911a9f42ebf372","target":"graph","created_at":"2026-05-18T04:27:57Z","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":"The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In partic","authors_text":"\\'Eric Colin de Verdi\\`ere, Gr\\'egory Ginot, Xavier Goaoc","cross_cats":["cs.CG","cs.DM","math.AT","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-31T16:21:38Z","title":"Helly numbers of acyclic families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.6006","kind":"arxiv","version":2},"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":"sha256:31be1da5838148326332667b37937fd1905434e501fde239a0c85152654839e4","target":"record","created_at":"2026-05-18T04:27:57Z","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":"0a96cbea9aee00c18e2cabcd7c45f248fc7db0543d6e5036c716aed85f41fa08","cross_cats_sorted":["cs.CG","cs.DM","math.AT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-31T16:21:38Z","title_canon_sha256":"327af055bc1ba250f3d14156245dd7e36c0540a2e7938922526383a4458b5b0d"},"schema_version":"1.0","source":{"id":"1101.6006","kind":"arxiv","version":2}},"canonical_sha256":"ffec824630cfaa232c112731db8eb0d4d55a40e1ff29c3a8d1eb2d5d78e0c138","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ffec824630cfaa232c112731db8eb0d4d55a40e1ff29c3a8d1eb2d5d78e0c138","first_computed_at":"2026-05-18T04:27:57.807705Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:57.807705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B7F4AqVl0DZ/7J1xx25xlWpA9T4f7lsyYjB51YiWo5+NJUQ2ynqCFLgJCjfUQ4TG9ue3WQpE7toy94Xiz4KtDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:57.808428Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.6006","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31be1da5838148326332667b37937fd1905434e501fde239a0c85152654839e4","sha256:72f3dfe98cbc5d6d02bad09b0ababaaefaec47304108b1c61e911a9f42ebf372"],"state_sha256":"d0b56b738580d56c660f9dc9e9ebfb29d05d7567f5706aaba1b351a7c84c9d94"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37dTlAVuKMUI6A+4I8JbCm135ZPPoftfEyE6oAWIBzI9dxERNMFbBv4aNuupX2kMndTSZIWlQk1NvYcZ9+EnBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T23:06:06.747445Z","bundle_sha256":"f0b42dd1e5b0127b3c7c7e05070ba0952b6b002cc997e310f24348aa353ee679"}}