{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:3JCGA4CHRE3BA2ZYMKTEMOO6NY","short_pith_number":"pith:3JCGA4CH","schema_version":"1.0","canonical_sha256":"da446070478936106b3862a64639de6e09274b8eeb13f85f05cf930c5423a883","source":{"kind":"arxiv","id":"1602.04763","version":3},"attestation_state":"computed","paper":{"title":"On the number of bases of almost all matroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jorn van der Pol, Rudi Pendavingh","submitted_at":"2016-02-15T18:50:27Z","abstract_excerpt":"For a matroid $M$ of rank $r$ on $n$ elements, let $b(M)$ denote the fraction of bases of $M$ among the subsets of the ground set with cardinality $r$. We show that $$\\Omega(1/n)\\leq 1-b(M)\\leq O(\\log(n)^3/n)\\text{ as }n\\rightarrow \\infty$$ for asymptotically almost all matroids $M$ on $n$ elements. We derive that asymptotically almost all matroids on $n$ elements (1) have a $U_{k,2k}$-minor, whenever $k\\leq O(\\log(n))$, (2) have girth $\\geq \\Omega(\\log(n))$, (3) have Tutte connectivity $\\geq \\Omega(\\sqrt{\\log(n)})$, and (4) do not arise as the truncation of another matroid.\n  Our argument is "},"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":"1602.04763","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-15T18:50:27Z","cross_cats_sorted":[],"title_canon_sha256":"0499b2b61b462295a81fc5078d064a0f3e056dc07dfd650a1f25224a5cf7796a","abstract_canon_sha256":"62b08d1c19aff4924a98d07753c8318f331e0a905e3b2b28217b8fee8dc2bcc0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.119624Z","signature_b64":"hItorWV6KQ83Vef6iyPs6iGg3nyfOKzV8fFCtu3MfKXfBNHSlnMMl7s5ffMOyoS95LsAl+DRdkkjces2e/8wAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da446070478936106b3862a64639de6e09274b8eeb13f85f05cf930c5423a883","last_reissued_at":"2026-05-18T01:01:42.118975Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.118975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of bases of almost all matroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jorn van der Pol, Rudi Pendavingh","submitted_at":"2016-02-15T18:50:27Z","abstract_excerpt":"For a matroid $M$ of rank $r$ on $n$ elements, let $b(M)$ denote the fraction of bases of $M$ among the subsets of the ground set with cardinality $r$. We show that $$\\Omega(1/n)\\leq 1-b(M)\\leq O(\\log(n)^3/n)\\text{ as }n\\rightarrow \\infty$$ for asymptotically almost all matroids $M$ on $n$ elements. We derive that asymptotically almost all matroids on $n$ elements (1) have a $U_{k,2k}$-minor, whenever $k\\leq O(\\log(n))$, (2) have girth $\\geq \\Omega(\\log(n))$, (3) have Tutte connectivity $\\geq \\Omega(\\sqrt{\\log(n)})$, and (4) do not arise as the truncation of another matroid.\n  Our argument is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04763","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":"1602.04763","created_at":"2026-05-18T01:01:42.119075+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.04763v3","created_at":"2026-05-18T01:01:42.119075+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04763","created_at":"2026-05-18T01:01:42.119075+00:00"},{"alias_kind":"pith_short_12","alias_value":"3JCGA4CHRE3B","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"3JCGA4CHRE3BA2ZY","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"3JCGA4CH","created_at":"2026-05-18T12:29:55.572404+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/3JCGA4CHRE3BA2ZYMKTEMOO6NY","json":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY.json","graph_json":"https://pith.science/api/pith-number/3JCGA4CHRE3BA2ZYMKTEMOO6NY/graph.json","events_json":"https://pith.science/api/pith-number/3JCGA4CHRE3BA2ZYMKTEMOO6NY/events.json","paper":"https://pith.science/paper/3JCGA4CH"},"agent_actions":{"view_html":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY","download_json":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY.json","view_paper":"https://pith.science/paper/3JCGA4CH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.04763&json=true","fetch_graph":"https://pith.science/api/pith-number/3JCGA4CHRE3BA2ZYMKTEMOO6NY/graph.json","fetch_events":"https://pith.science/api/pith-number/3JCGA4CHRE3BA2ZYMKTEMOO6NY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY/action/storage_attestation","attest_author":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY/action/author_attestation","sign_citation":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY/action/citation_signature","submit_replication":"https://pith.science/pith/3JCGA4CHRE3BA2ZYMKTEMOO6NY/action/replication_record"}},"created_at":"2026-05-18T01:01:42.119075+00:00","updated_at":"2026-05-18T01:01:42.119075+00:00"}