{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:XQKLKYGOFOXCSJ7CU6DUZOZWT4","short_pith_number":"pith:XQKLKYGO","canonical_record":{"source":{"id":"2607.23928","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-27T01:48:38Z","cross_cats_sorted":[],"title_canon_sha256":"d2f7b66d313d9df5e745e3bd405b8e82489e037e5860db984346c6370e3c997a","abstract_canon_sha256":"061d0a85034ffc74837070529c4be5c94c70dea1c32739c22b399bb153a0b20a"},"schema_version":"1.0"},"canonical_sha256":"bc14b560ce2bae2927e2a7874cbb369f3cc41062b8b1e7ee6f4bfd84c18221c2","source":{"kind":"arxiv","id":"2607.23928","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.23928","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.23928v1","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23928","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_12","alias_value":"XQKLKYGOFOXC","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_16","alias_value":"XQKLKYGOFOXCSJ7C","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_8","alias_value":"XQKLKYGO","created_at":"2026-07-28T01:23:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:XQKLKYGOFOXCSJ7CU6DUZOZWT4","target":"record","payload":{"canonical_record":{"source":{"id":"2607.23928","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-27T01:48:38Z","cross_cats_sorted":[],"title_canon_sha256":"d2f7b66d313d9df5e745e3bd405b8e82489e037e5860db984346c6370e3c997a","abstract_canon_sha256":"061d0a85034ffc74837070529c4be5c94c70dea1c32739c22b399bb153a0b20a"},"schema_version":"1.0"},"canonical_sha256":"bc14b560ce2bae2927e2a7874cbb369f3cc41062b8b1e7ee6f4bfd84c18221c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T01:23:18.137718Z","signature_b64":"A23BOCQf8zGM40aEGJnjLMvb+ocvwnWdjrSDl6ScMHWfYpDMhK4UVEg0/ATpqr+IsHnBrJo2lwL99JF5wcmhAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc14b560ce2bae2927e2a7874cbb369f3cc41062b8b1e7ee6f4bfd84c18221c2","last_reissued_at":"2026-07-28T01:23:18.136858Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T01:23:18.136858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.23928","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":"2026-07-28T01:23:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QzEIxGwChbo9fZlkPgDxmTAB2+BayVckk8Zr9reosjuPRWrw7uCr41fkRa8uAuaiT0umoZ2YRa6w2lBritaBDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T02:21:02.032672Z"},"content_sha256":"d530ac0377f9c8bafa55afd573baa273a8ab6ce996671264a453b5ecdc715f80","schema_version":"1.0","event_id":"sha256:d530ac0377f9c8bafa55afd573baa273a8ab6ce996671264a453b5ecdc715f80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:XQKLKYGOFOXCSJ7CU6DUZOZWT4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on matchings and co-matchings in bipartite graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sida Li","submitted_at":"2026-07-27T01:48:38Z","abstract_excerpt":"A class of bipartite graphs is said to have the strong Erd\\H{o}s-Hajnal property if there exists $\\varepsilon > 0$ such that every graph $((A, B), E)$ in the class contains a complete or empty induced subgraph with parts $X \\subseteq A$, $Y \\subseteq B$ where $|X| \\ge \\varepsilon|A|$ and $|Y| \\ge \\varepsilon|B|$. Scott, Seymour and Spirkl \\cite{scott2023} proved that it is enough to forbid a forest and the bipartite complement of a forest. In this paper, we provide quantitative bounds on $\\varepsilon$ when we restrict to matchings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23928","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.23928/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":"2026-07-28T01:23:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TTEnKN4yJwv1EuMl4I7sLoUTjRdBVGOMvZGKtOGBqbfmxZns3mrC5zahCNtQo4RxWqzin/bV2WHWv2udrOCaBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T02:21:02.033197Z"},"content_sha256":"8f7143f02eec5aa3d39c71fd5711d7043f59a2835f0ebbe39e76a5078d61487f","schema_version":"1.0","event_id":"sha256:8f7143f02eec5aa3d39c71fd5711d7043f59a2835f0ebbe39e76a5078d61487f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/bundle.json","state_url":"https://pith.science/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/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":"2026-08-09T02:21:02Z","links":{"resolver":"https://pith.science/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4","bundle":"https://pith.science/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/bundle.json","state":"https://pith.science/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XQKLKYGOFOXCSJ7CU6DUZOZWT4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:XQKLKYGOFOXCSJ7CU6DUZOZWT4","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":"061d0a85034ffc74837070529c4be5c94c70dea1c32739c22b399bb153a0b20a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-27T01:48:38Z","title_canon_sha256":"d2f7b66d313d9df5e745e3bd405b8e82489e037e5860db984346c6370e3c997a"},"schema_version":"1.0","source":{"id":"2607.23928","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.23928","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.23928v1","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23928","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_12","alias_value":"XQKLKYGOFOXC","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_16","alias_value":"XQKLKYGOFOXCSJ7C","created_at":"2026-07-28T01:23:18Z"},{"alias_kind":"pith_short_8","alias_value":"XQKLKYGO","created_at":"2026-07-28T01:23:18Z"}],"graph_snapshots":[{"event_id":"sha256:8f7143f02eec5aa3d39c71fd5711d7043f59a2835f0ebbe39e76a5078d61487f","target":"graph","created_at":"2026-07-28T01:23:18Z","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/2607.23928/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A class of bipartite graphs is said to have the strong Erd\\H{o}s-Hajnal property if there exists $\\varepsilon > 0$ such that every graph $((A, B), E)$ in the class contains a complete or empty induced subgraph with parts $X \\subseteq A$, $Y \\subseteq B$ where $|X| \\ge \\varepsilon|A|$ and $|Y| \\ge \\varepsilon|B|$. Scott, Seymour and Spirkl \\cite{scott2023} proved that it is enough to forbid a forest and the bipartite complement of a forest. In this paper, we provide quantitative bounds on $\\varepsilon$ when we restrict to matchings.","authors_text":"Sida Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23928","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:d530ac0377f9c8bafa55afd573baa273a8ab6ce996671264a453b5ecdc715f80","target":"record","created_at":"2026-07-28T01:23:18Z","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":"061d0a85034ffc74837070529c4be5c94c70dea1c32739c22b399bb153a0b20a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-27T01:48:38Z","title_canon_sha256":"d2f7b66d313d9df5e745e3bd405b8e82489e037e5860db984346c6370e3c997a"},"schema_version":"1.0","source":{"id":"2607.23928","kind":"arxiv","version":1}},"canonical_sha256":"bc14b560ce2bae2927e2a7874cbb369f3cc41062b8b1e7ee6f4bfd84c18221c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc14b560ce2bae2927e2a7874cbb369f3cc41062b8b1e7ee6f4bfd84c18221c2","first_computed_at":"2026-07-28T01:23:18.136858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-28T01:23:18.136858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A23BOCQf8zGM40aEGJnjLMvb+ocvwnWdjrSDl6ScMHWfYpDMhK4UVEg0/ATpqr+IsHnBrJo2lwL99JF5wcmhAg==","signature_status":"signed_v1","signed_at":"2026-07-28T01:23:18.137718Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.23928","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d530ac0377f9c8bafa55afd573baa273a8ab6ce996671264a453b5ecdc715f80","sha256:8f7143f02eec5aa3d39c71fd5711d7043f59a2835f0ebbe39e76a5078d61487f"],"state_sha256":"7da7b6a2cec797f414b13866922117a58416737ec04b0ea5d392d95b9739762f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mbyf9bPY+IqvTDky12NEp5K3r8SfSzPjvz9P5IRQnokkfYFDgUPsZf8Uesb6iXrHiNVxPwgJ+ninxLWwKp6zCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T02:21:02.037986Z","bundle_sha256":"492c520b9bd78953916dc17e390ebfc0952079cd9ae8a459868f8ce296b27be4"}}