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Aharoni and Berger \\cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \\ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \\cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \\cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\\sqrt{n}$."},"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":"1405.3119","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:07:25Z","cross_cats_sorted":[],"title_canon_sha256":"acd40548e0d6164a5a316135694b554b0843f22d63ac155acdbb70cab6506e16","abstract_canon_sha256":"fcadf706ec7e8a467eba3fc0eeef8eedcf68f0c756813f74cfd21074d954e2db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:16.648400Z","signature_b64":"uD9Ki0aGzZfih3ZC1CY/VV8f/Ft7KXfNuPsRq9CktypkjPGM3uv/3FAylfVdG9yVnTYWQMfOR0I7Gj+B/g6YDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bcc16210098317b66ec909a4178fc7327424eaeff730ee8ea59e2e84276ae06","last_reissued_at":"2026-05-18T01:35:16.647712Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:16.647712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rainbow sets in the intersection of two matroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Kotlar, Ran Ziv, Ron Aharoni","submitted_at":"2014-05-13T12:07:25Z","abstract_excerpt":"Given sets $F_1, \\ldots ,F_n$, a {\\em partial rainbow function} is a partial choice function of the sets $F_i$. 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Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\\sqrt{n}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3119","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":""},"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":"1405.3119","created_at":"2026-05-18T01:35:16.647821+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.3119v3","created_at":"2026-05-18T01:35:16.647821+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.3119","created_at":"2026-05-18T01:35:16.647821+00:00"},{"alias_kind":"pith_short_12","alias_value":"TPGBMIIATAYX","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"TPGBMIIATAYXWZXM","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"TPGBMIIA","created_at":"2026-05-18T12:28:49.207871+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/TPGBMIIATAYXWZXMSCNEC6H4OM","json":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM.json","graph_json":"https://pith.science/api/pith-number/TPGBMIIATAYXWZXMSCNEC6H4OM/graph.json","events_json":"https://pith.science/api/pith-number/TPGBMIIATAYXWZXMSCNEC6H4OM/events.json","paper":"https://pith.science/paper/TPGBMIIA"},"agent_actions":{"view_html":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM","download_json":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM.json","view_paper":"https://pith.science/paper/TPGBMIIA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.3119&json=true","fetch_graph":"https://pith.science/api/pith-number/TPGBMIIATAYXWZXMSCNEC6H4OM/graph.json","fetch_events":"https://pith.science/api/pith-number/TPGBMIIATAYXWZXMSCNEC6H4OM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM/action/storage_attestation","attest_author":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM/action/author_attestation","sign_citation":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM/action/citation_signature","submit_replication":"https://pith.science/pith/TPGBMIIATAYXWZXMSCNEC6H4OM/action/replication_record"}},"created_at":"2026-05-18T01:35:16.647821+00:00","updated_at":"2026-05-18T01:35:16.647821+00:00"}