{"as_of":"2026-08-09T06:05:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:1a8c0da91a5d75d75fa79e77984a1afad34ba134fd7dd40e04f37913acfbda90","coverage":[{"denominator":17,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":17,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T04:15:23.978196Z","state":"measured"},{"denominator":17,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":17,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-08T06:32:00.761636+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2607.23928/citation-record","integrity":"/paper/2607.23928/integrity","json":"/paper/2607.23928/citation-record.json","paper":"/paper/2607.23928"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.392947Z","title":"Crossing patterns of semi-algebraic sets.Journal of Combinatorial Theory, Series A, 111(2):310–326, 2005","venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.392947Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:eca8bacec30672add0e77dacb04d4a1334b6ef40b9ed2ea8db6aaa0e8c5aa022","observation_id":"64ce8e2d-d9b5-4fb4-b4e5-397dc566df60","resolution":{"observed_at":"2026-08-04T04:15:22.392947Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.23989","last_updated":"2025-06-30T15:54:45Z","snapshot_observed_at":"2026-08-06T21:24:25.908420Z","submitted_at":"2025-06-30T15:54:45Z","title":"Factorization norms and an inverse theorem for MaxCut","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.23989","snapshot_observed_at":"2026-08-04T04:15:22.467675Z","title":"Factorization norms and an inverse theorem for MaxCut, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.467675Z"},"links":{"cited_paper":"/paper/2506.23989","citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:4f9a33095cdb8b4d4b4d9d0c5133a43fd1858809e5c00cd33659e0a66dc805a5","observation_id":"1f348c96-7725-4473-8025-c91e916fe92e","resolution":{"observed_at":"2026-08-04T04:15:22.467675Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.636369Z","title":"Pure pairs","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.636369Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:1a0167035c3d20826a70d9a10676549a61b0de6bf0ed80c01b8429f264602810","observation_id":"b9773393-2193-4a97-9613-d09255b81edb","resolution":{"observed_at":"2026-08-04T04:15:22.636369Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.745614Z","title":"Ramsey-type theorems.Discrete Applied Mathematics, 25(1):37–52, 1989","venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.745614Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:1f06d7951772979fc3ea193b636a296d20f603c18d2d6278785edd181ffdf45a","observation_id":"f7f2756f-8c2c-4aeb-8819-e80e4a5269ce","resolution":{"observed_at":"2026-08-04T04:15:22.745614Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.846211Z","title":"Springer Berlin Heidelberg, Berlin, Heidelberg, 2008","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.846211Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:32fe2ec15e60a6e1137e154f00c15c3f579a8b9061537e46edd16cb98afb1911","observation_id":"d3015339-f209-4930-89e3-d7861a71a5c9","resolution":{"observed_at":"2026-08-04T04:15:22.846211Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.859055Z","title":"En route to the log-rank conjecture: New reductions and equivalent formulations","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.859055Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:c09b98952121f68c8a1406d048048ce3d97ca2d6921313cbf8767fe5e485b29d","observation_id":"54546f34-db2b-4b75-914f-d71e8329b5cc","resolution":{"observed_at":"2026-08-04T04:15:22.859055Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:22.995801Z","title":"Deterministic communication vs","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:22.995801Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:cc6dd4a845ddbf33726befcd6cbc62ceb81d6e77539b02dacb6ca0bf4560b6bf","observation_id":"d2ddd066-be15-435e-8ef1-d7de9ca89ce4","resolution":{"observed_at":"2026-08-04T04:15:22.995801Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.114193Z","title":"Geometric rank of tensors and subrank of matrix multiplication.Discrete Analysis, 2023","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.114193Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:fa00b246795d8e90eaafda69568f6390586bb8d12bbc22edec4fac01417decd1","observation_id":"51e40843-b81f-4108-b152-e8783cbb6149","resolution":{"observed_at":"2026-08-04T04:15:23.114193Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.228880Z","title":"Communication is bounded by root of rank.J","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.228880Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:45ad39e66051c077be5d84b37dbff14980cb32fa68f45df2a192fc22dfd9f44d","observation_id":"d69e94d0-f0ee-4d07-881f-6292aa80ef44","resolution":{"observed_at":"2026-08-04T04:15:23.228880Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.352042Z","title":"Lattices, mobius functions and communications complexity","venue":null,"work_id":null,"year":1988},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.352042Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:3f7617dcd703a1724b68d8a8d087de8d8b9450c2b6c6175788156309c02920b4","observation_id":"12ef8bc2-e680-4627-a1cd-290f1c216b20","resolution":{"observed_at":"2026-08-04T04:15:23.352042Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.474092Z","title":"The partition rank of a tensor andk-right corners inF qn.Journal of Combina- torial Theory, Series A, 174:105190, 2020","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.474092Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:7a11f06bcde972d64c82bcefa52e0067e5920ac46526a97f669fc2e2f7650bad","observation_id":"a843b368-e435-4d3a-b5fb-3c4d0f4055d5","resolution":{"observed_at":"2026-08-04T04:15:23.474092Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.561088Z","title":"On rank vs","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.561088Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:d56aba3097b42c9afc983b1f7a54bea91434ee4673e76a20706fa60eeb7561c9","observation_id":"736709b9-f1d5-484e-a32c-6afaf62253d3","resolution":{"observed_at":"2026-08-04T04:15:23.561088Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.633875Z","title":"Payne and Joseph A","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.633875Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:01f2af1f7768543db111325bd31645421f2c342ca5bf130f25f06ac791d7e081","observation_id":"12197fca-2df1-48b0-b3c4-e2f6fa4ce931","resolution":{"observed_at":"2026-08-04T04:15:23.633875Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.719146Z","title":"Pure pairs","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.719146Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:e379dc0c12e2cb5cb1b24e467bd00e4dbc818c7e5d7c4aef40e2d41b94a30200","observation_id":"12d2f624-95fb-40f6-aae1-9b8494fce8c5","resolution":{"observed_at":"2026-08-04T04:15:23.719146Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.792952Z","title":"Matrix discrepancy and the log-rank conjecture.Math","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.792952Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:99bd271bc7004515ae27fb023eebe607688780247967ee4bd06cd9a2073ce208","observation_id":"f8f0d36c-4a35-43f9-9f21-8529555996ad","resolution":{"observed_at":"2026-08-04T04:15:23.792952Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.903321Z","title":"A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound,","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.903321Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:ec47ac6199de2b01e99ca50317fc6457962dbd418ee6a6aa172616412ab8f13c","observation_id":"f7a58618-76d2-4025-bc99-df546d322eaf","resolution":{"observed_at":"2026-08-04T04:15:23.903321Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T04:15:23.978196Z","title":null,"venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs","version":2},"reference_index":2016,"source":"pdf_text","source_observed_at":"2026-08-04T04:15:23.978196Z"},"links":{"citing_paper":"/paper/2607.23928"},"observation_digest":"sha256:06b108faaaf70a7e32f0b787e4c55998a177ee877f784fa5dd2b4f7c5a3afcf9","observation_id":"f1625e4e-41bf-4bb2-9eb8-023a90b1e685","resolution":{"observed_at":"2026-08-04T04:15:23.978196Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2607.23928","last_updated":"2026-08-03T15:34:06Z","latest_version":2,"primary_category":"math.CO","snapshot_observed_at":"2026-08-06T23:37:51.544864Z","submitted_at":"2026-07-27T01:48:38Z","title":"A note on matchings and co-matchings in bipartite graphs"},"reference_resolution":{"displayed":17,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":17,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":17},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+00:00","source":"retraction_watch"}],"thesis":"As of 9 August 2026, this Paper Citation Record lists 17 of 17 outbound references and 0 inbound Pith citation observations for arXiv:2607.23928."}