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Clemens and Ehrenm{\\\"u}ller proved that $f(n) \\le \\frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \\le \\lceil \\frac{3}{2}n \\rceil+1$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.00943","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-05T19:27:36Z","cross_cats_sorted":[],"title_canon_sha256":"8bd987750ce52c85b3710cb5ad516468a9aa69c55bc3f7a670a5c03894f716e9","abstract_canon_sha256":"ca175df69c52794ca6c59755fcf987627289c0e9c82ca694d443be74b48ec937"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:09.811480Z","signature_b64":"MP7Vwtihd92erbNg2Bk2Mr7/yKRoe4/LUvizmo2m4/rBSkVpv1WbRLzcqccJll0/TNtsXsFFdjVO7tr0kB/iAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6600154c40cfb6f941b60c05d00329172d269f6e31a498b1c29a3f6a74d243a4","last_reissued_at":"2026-05-18T00:50:09.810950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:09.810950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representation of large matchings in bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dani Kotlar, Ran Ziv, Ron Aharoni","submitted_at":"2016-01-05T19:27:36Z","abstract_excerpt":"Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. 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We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \\le \\lceil \\frac{3}{2}n \\rceil+1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00943","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1601.00943","created_at":"2026-05-18T00:50:09.811025+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.00943v2","created_at":"2026-05-18T00:50:09.811025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00943","created_at":"2026-05-18T00:50:09.811025+00:00"},{"alias_kind":"pith_short_12","alias_value":"MYABKTCAZ63P","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MYABKTCAZ63PSQNW","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MYABKTCA","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4","json":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4.json","graph_json":"https://pith.science/api/pith-number/MYABKTCAZ63PSQNWBQC5AAZJC4/graph.json","events_json":"https://pith.science/api/pith-number/MYABKTCAZ63PSQNWBQC5AAZJC4/events.json","paper":"https://pith.science/paper/MYABKTCA"},"agent_actions":{"view_html":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4","download_json":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4.json","view_paper":"https://pith.science/paper/MYABKTCA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.00943&json=true","fetch_graph":"https://pith.science/api/pith-number/MYABKTCAZ63PSQNWBQC5AAZJC4/graph.json","fetch_events":"https://pith.science/api/pith-number/MYABKTCAZ63PSQNWBQC5AAZJC4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4/action/storage_attestation","attest_author":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4/action/author_attestation","sign_citation":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4/action/citation_signature","submit_replication":"https://pith.science/pith/MYABKTCAZ63PSQNWBQC5AAZJC4/action/replication_record"}},"created_at":"2026-05-18T00:50:09.811025+00:00","updated_at":"2026-05-18T00:50:09.811025+00:00"}