{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:EHWSV2APV7F5FEYEQE26DFXKPA","short_pith_number":"pith:EHWSV2AP","canonical_record":{"source":{"id":"2012.14992","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-12-30T00:55:59Z","cross_cats_sorted":[],"title_canon_sha256":"019cd7465ed9841a3a1a5f33dd65083dde6c26a1d44fef46f50885a772d5116d","abstract_canon_sha256":"a4153f8ba29d2870a215e585bbbf3b06466ba186b26963c3a4c4c450debe9f9c"},"schema_version":"1.0"},"canonical_sha256":"21ed2ae80fafcbd293048135e196ea781c57d64b3929dabaa01dc97f9c47224b","source":{"kind":"arxiv","id":"2012.14992","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2012.14992","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"arxiv_version","alias_value":"2012.14992v3","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.14992","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_12","alias_value":"EHWSV2APV7F5","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_16","alias_value":"EHWSV2APV7F5FEYE","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_8","alias_value":"EHWSV2AP","created_at":"2026-07-05T03:20:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:EHWSV2APV7F5FEYEQE26DFXKPA","target":"record","payload":{"canonical_record":{"source":{"id":"2012.14992","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-12-30T00:55:59Z","cross_cats_sorted":[],"title_canon_sha256":"019cd7465ed9841a3a1a5f33dd65083dde6c26a1d44fef46f50885a772d5116d","abstract_canon_sha256":"a4153f8ba29d2870a215e585bbbf3b06466ba186b26963c3a4c4c450debe9f9c"},"schema_version":"1.0"},"canonical_sha256":"21ed2ae80fafcbd293048135e196ea781c57d64b3929dabaa01dc97f9c47224b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:20:33.691985Z","signature_b64":"TaKRUqRiqKFaBILytPfqh0ogmj7i2jAbU+p1CePhqqPx52J0bqCcj/O4a7KRErc2O1AzsXNO/X1xN58PRcv/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21ed2ae80fafcbd293048135e196ea781c57d64b3929dabaa01dc97f9c47224b","last_reissued_at":"2026-07-05T03:20:33.691580Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:20:33.691580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2012.14992","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:20:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qC47Fnae55sl5IuN+1ftdg/IvQmqAVVsODWrxb2sUSRWy2D38srrhsgIfRoSMsCGGzcp2AD5xK+TqhpCkJKDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:28:10.706754Z"},"content_sha256":"86fa9a0e3bbb6a58da70289e5461fe72b77d18e7e41fa9b3bf31701327b1926b","schema_version":"1.0","event_id":"sha256:86fa9a0e3bbb6a58da70289e5461fe72b77d18e7e41fa9b3bf31701327b1926b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:EHWSV2APV7F5FEYEQE26DFXKPA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rainbow paths and large rainbow matchings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eli Berger, Maria Chudnovsky, Ron Aharoni, Shira Zerbib","submitted_at":"2020-12-30T00:55:59Z","abstract_excerpt":"A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\\frac{2}{3}n-1$, improving on the trivial $\\frac{1}{2}n$, and an analogous result for hypergraphs. For $\\{C_3,C_5\\}$-free graphs and for disjoint matchings we obtain a lower bound of $\\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.14992","kind":"arxiv","version":3},"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/2012.14992/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:20:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"18TyWo2F6eC68ZHlDogMxWnbv0evEPQWVmmAGXcIOCihNVq3tn/+no6mVaWQoPmGK7N/pWtOrE5RkqsczyvWDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:28:10.707168Z"},"content_sha256":"5b7cb3edb4acc49b759a2f8a2b0cf16c148748074a1674b7f9f09053cb010a92","schema_version":"1.0","event_id":"sha256:5b7cb3edb4acc49b759a2f8a2b0cf16c148748074a1674b7f9f09053cb010a92"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EHWSV2APV7F5FEYEQE26DFXKPA/bundle.json","state_url":"https://pith.science/pith/EHWSV2APV7F5FEYEQE26DFXKPA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EHWSV2APV7F5FEYEQE26DFXKPA/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":"2026-07-07T13:28:10Z","links":{"resolver":"https://pith.science/pith/EHWSV2APV7F5FEYEQE26DFXKPA","bundle":"https://pith.science/pith/EHWSV2APV7F5FEYEQE26DFXKPA/bundle.json","state":"https://pith.science/pith/EHWSV2APV7F5FEYEQE26DFXKPA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EHWSV2APV7F5FEYEQE26DFXKPA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:EHWSV2APV7F5FEYEQE26DFXKPA","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":"a4153f8ba29d2870a215e585bbbf3b06466ba186b26963c3a4c4c450debe9f9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-12-30T00:55:59Z","title_canon_sha256":"019cd7465ed9841a3a1a5f33dd65083dde6c26a1d44fef46f50885a772d5116d"},"schema_version":"1.0","source":{"id":"2012.14992","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2012.14992","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"arxiv_version","alias_value":"2012.14992v3","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.14992","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_12","alias_value":"EHWSV2APV7F5","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_16","alias_value":"EHWSV2APV7F5FEYE","created_at":"2026-07-05T03:20:33Z"},{"alias_kind":"pith_short_8","alias_value":"EHWSV2AP","created_at":"2026-07-05T03:20:33Z"}],"graph_snapshots":[{"event_id":"sha256:5b7cb3edb4acc49b759a2f8a2b0cf16c148748074a1674b7f9f09053cb010a92","target":"graph","created_at":"2026-07-05T03:20:33Z","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/2012.14992/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\\frac{2}{3}n-1$, improving on the trivial $\\frac{1}{2}n$, and an analogous result for hypergraphs. For $\\{C_3,C_5\\}$-free graphs and for disjoint matchings we obtain a lower bound of $\\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture.","authors_text":"Eli Berger, Maria Chudnovsky, Ron Aharoni, Shira Zerbib","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-12-30T00:55:59Z","title":"Rainbow paths and large rainbow matchings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.14992","kind":"arxiv","version":3},"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":"sha256:86fa9a0e3bbb6a58da70289e5461fe72b77d18e7e41fa9b3bf31701327b1926b","target":"record","created_at":"2026-07-05T03:20:33Z","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":"a4153f8ba29d2870a215e585bbbf3b06466ba186b26963c3a4c4c450debe9f9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-12-30T00:55:59Z","title_canon_sha256":"019cd7465ed9841a3a1a5f33dd65083dde6c26a1d44fef46f50885a772d5116d"},"schema_version":"1.0","source":{"id":"2012.14992","kind":"arxiv","version":3}},"canonical_sha256":"21ed2ae80fafcbd293048135e196ea781c57d64b3929dabaa01dc97f9c47224b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21ed2ae80fafcbd293048135e196ea781c57d64b3929dabaa01dc97f9c47224b","first_computed_at":"2026-07-05T03:20:33.691580Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:20:33.691580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TaKRUqRiqKFaBILytPfqh0ogmj7i2jAbU+p1CePhqqPx52J0bqCcj/O4a7KRErc2O1AzsXNO/X1xN58PRcv/DA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:20:33.691985Z","signed_message":"canonical_sha256_bytes"},"source_id":"2012.14992","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86fa9a0e3bbb6a58da70289e5461fe72b77d18e7e41fa9b3bf31701327b1926b","sha256:5b7cb3edb4acc49b759a2f8a2b0cf16c148748074a1674b7f9f09053cb010a92"],"state_sha256":"351b10b2e0934fc5db8ddf9b1332583f32a1fc52a93b15de9b57e0d595750a80"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FBK019xmXWKxj6qhF33AkKVAOf7TPhM7Kn/yLPmVNd32SKL+9r54h/zRvtLvfhMgrmvJ9lOaNRC5IuvQU93tDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T13:28:10.709029Z","bundle_sha256":"52c572a3291051631ca7d34754a90058e79835dfd3edd2c3a5c7d76638bd35ac"}}