{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WQTKOJMZH5N3QIIWEFU6U54HUJ","short_pith_number":"pith:WQTKOJMZ","canonical_record":{"source":{"id":"1210.7359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-27T18:47:52Z","cross_cats_sorted":[],"title_canon_sha256":"2e5161efadcc50ab3feee01b4820e55a39c8c2471222dd30f761180c3c12734d","abstract_canon_sha256":"72fe2aefede641d3a9c8cf553bef4ae34ba96aeb249774813f8ca7957ccec4e0"},"schema_version":"1.0"},"canonical_sha256":"b426a725993f5bb821162169ea7787a254244cc741a10a07fc96913f3f5d1e30","source":{"kind":"arxiv","id":"1210.7359","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7359","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7359v1","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7359","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"pith_short_12","alias_value":"WQTKOJMZH5N3","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WQTKOJMZH5N3QIIW","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WQTKOJMZ","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WQTKOJMZH5N3QIIWEFU6U54HUJ","target":"record","payload":{"canonical_record":{"source":{"id":"1210.7359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-27T18:47:52Z","cross_cats_sorted":[],"title_canon_sha256":"2e5161efadcc50ab3feee01b4820e55a39c8c2471222dd30f761180c3c12734d","abstract_canon_sha256":"72fe2aefede641d3a9c8cf553bef4ae34ba96aeb249774813f8ca7957ccec4e0"},"schema_version":"1.0"},"canonical_sha256":"b426a725993f5bb821162169ea7787a254244cc741a10a07fc96913f3f5d1e30","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:10.296470Z","signature_b64":"0+mn6mz8hJtuRqMvl1mScWgWBcLnACgpa6+k6nqVW5vmDI8n6THw12r79mN1n2k5UdUo99uZhsYFik53HftQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b426a725993f5bb821162169ea7787a254244cc741a10a07fc96913f3f5d1e30","last_reissued_at":"2026-05-18T03:42:10.295662Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:10.295662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.7359","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DnjNiedYlVml0dT5qN8dq9g7LsmfdZVLsbftu5zs4oA13LO5JveYE041tuHXHqhlhuJ21YwkPBjAy42y+dU+AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:47:13.136463Z"},"content_sha256":"feb7bae080e73d3cc7a07af178cedc35055fa33b2a8e081edfa86f70b12d91b0","schema_version":"1.0","event_id":"sha256:feb7bae080e73d3cc7a07af178cedc35055fa33b2a8e081edfa86f70b12d91b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WQTKOJMZH5N3QIIWEFU6U54HUJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Treglown, Yi Zhao","submitted_at":"2012-10-27T18:47:52Z","abstract_excerpt":"Given positive integers k\\geq 3 and r where k/2 \\leq r \\leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7359","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":""},"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":"2026-05-18T03:42:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CNahjBsBfeKvo6zusOWt+bMdTNu+AcjVYcUuB5cF5bMzls/zkfPyWU+GY/wcdumratLIEEoRSvENMfAFQwbiCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:47:13.136814Z"},"content_sha256":"f3e76f7ede6cf2f05f920ba082798ecd38e688f6019ed6e30ed2ab58fcfe9aac","schema_version":"1.0","event_id":"sha256:f3e76f7ede6cf2f05f920ba082798ecd38e688f6019ed6e30ed2ab58fcfe9aac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/bundle.json","state_url":"https://pith.science/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/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":"2026-06-22T22:47:13Z","links":{"resolver":"https://pith.science/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ","bundle":"https://pith.science/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/bundle.json","state":"https://pith.science/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WQTKOJMZH5N3QIIWEFU6U54HUJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WQTKOJMZH5N3QIIWEFU6U54HUJ","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":"72fe2aefede641d3a9c8cf553bef4ae34ba96aeb249774813f8ca7957ccec4e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-27T18:47:52Z","title_canon_sha256":"2e5161efadcc50ab3feee01b4820e55a39c8c2471222dd30f761180c3c12734d"},"schema_version":"1.0","source":{"id":"1210.7359","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7359","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7359v1","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7359","created_at":"2026-05-18T03:42:10Z"},{"alias_kind":"pith_short_12","alias_value":"WQTKOJMZH5N3","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WQTKOJMZH5N3QIIW","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WQTKOJMZ","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:f3e76f7ede6cf2f05f920ba082798ecd38e688f6019ed6e30ed2ab58fcfe9aac","target":"graph","created_at":"2026-05-18T03:42:10Z","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":"Given positive integers k\\geq 3 and r where k/2 \\leq r \\leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4.","authors_text":"Andrew Treglown, Yi Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-27T18:47:52Z","title":"Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7359","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:feb7bae080e73d3cc7a07af178cedc35055fa33b2a8e081edfa86f70b12d91b0","target":"record","created_at":"2026-05-18T03:42:10Z","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":"72fe2aefede641d3a9c8cf553bef4ae34ba96aeb249774813f8ca7957ccec4e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-27T18:47:52Z","title_canon_sha256":"2e5161efadcc50ab3feee01b4820e55a39c8c2471222dd30f761180c3c12734d"},"schema_version":"1.0","source":{"id":"1210.7359","kind":"arxiv","version":1}},"canonical_sha256":"b426a725993f5bb821162169ea7787a254244cc741a10a07fc96913f3f5d1e30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b426a725993f5bb821162169ea7787a254244cc741a10a07fc96913f3f5d1e30","first_computed_at":"2026-05-18T03:42:10.295662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:10.295662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0+mn6mz8hJtuRqMvl1mScWgWBcLnACgpa6+k6nqVW5vmDI8n6THw12r79mN1n2k5UdUo99uZhsYFik53HftQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:10.296470Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.7359","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:feb7bae080e73d3cc7a07af178cedc35055fa33b2a8e081edfa86f70b12d91b0","sha256:f3e76f7ede6cf2f05f920ba082798ecd38e688f6019ed6e30ed2ab58fcfe9aac"],"state_sha256":"a39037c47200ee58210aaea868f1cf058c4e030e60f885d9838fe3fa859df532"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f4+1gj0/py5UBBoKlpfwELTpr1M/FlVxYFkalfMbJqXhw3lZbWdWBCZn+YvuI92w3tFZnWTFdAOCoz8HsnZXDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T22:47:13.138733Z","bundle_sha256":"6811d60036bfe52723d70627010bea287803adeb9ef6712ba8424ac1f09b21b9"}}