{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:5OZ7KHDXW5WOVF2PWXDQHFZYIS","short_pith_number":"pith:5OZ7KHDX","schema_version":"1.0","canonical_sha256":"ebb3f51c77b76cea974fb5c7039738449fba5893a61757e3c1ed9834576bb283","source":{"kind":"arxiv","id":"1908.01267","version":2},"attestation_state":"computed","paper":{"title":"Most binary matrices have no small defining set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anita Liebenau, Carly Bodkin, Ian M. Wanless","submitted_at":"2019-08-04T03:43:07Z","abstract_excerpt":"Consider a matrix $M$ chosen uniformly at random from a class of $m \\times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\\mathit{defining}$ $\\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\\mathit{size}$ of a defining set is the number of filled entries. A $\\mathit{critical}$ $\\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $\\epsilon>0$, we assume that $n\\le m=o(n^{1+\\epsilon})$, and that $\\lambda\\le1/2$, where $\\l"},"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":"1908.01267","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-04T03:43:07Z","cross_cats_sorted":[],"title_canon_sha256":"48b9c7ae998e784bd982422ffc00905f00c8cacf731fd7c20abe1a185022f69c","abstract_canon_sha256":"e7d85c2f7b7051d04243042f2708273d3bfbc242bf04fb44faa1c3bba7d4edc0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:13:23.310602Z","signature_b64":"mAsOZOjpl26NKmiK09bGNGTnCAaVU7K5Ak18hPSkcVVXes0Kb7mP3DnIzrbJVXYwDW11tEnPEqzZC+Q9dKs2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebb3f51c77b76cea974fb5c7039738449fba5893a61757e3c1ed9834576bb283","last_reissued_at":"2026-07-05T01:13:23.310193Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:13:23.310193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Most binary matrices have no small defining set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anita Liebenau, Carly Bodkin, Ian M. Wanless","submitted_at":"2019-08-04T03:43:07Z","abstract_excerpt":"Consider a matrix $M$ chosen uniformly at random from a class of $m \\times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\\mathit{defining}$ $\\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\\mathit{size}$ of a defining set is the number of filled entries. A $\\mathit{critical}$ $\\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $\\epsilon>0$, we assume that $n\\le m=o(n^{1+\\epsilon})$, and that $\\lambda\\le1/2$, where $\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01267","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01267/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.01267","created_at":"2026-07-05T01:13:23.310248+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.01267v2","created_at":"2026-07-05T01:13:23.310248+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01267","created_at":"2026-07-05T01:13:23.310248+00:00"},{"alias_kind":"pith_short_12","alias_value":"5OZ7KHDXW5WO","created_at":"2026-07-05T01:13:23.310248+00:00"},{"alias_kind":"pith_short_16","alias_value":"5OZ7KHDXW5WOVF2P","created_at":"2026-07-05T01:13:23.310248+00:00"},{"alias_kind":"pith_short_8","alias_value":"5OZ7KHDX","created_at":"2026-07-05T01:13:23.310248+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/5OZ7KHDXW5WOVF2PWXDQHFZYIS","json":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS.json","graph_json":"https://pith.science/api/pith-number/5OZ7KHDXW5WOVF2PWXDQHFZYIS/graph.json","events_json":"https://pith.science/api/pith-number/5OZ7KHDXW5WOVF2PWXDQHFZYIS/events.json","paper":"https://pith.science/paper/5OZ7KHDX"},"agent_actions":{"view_html":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS","download_json":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS.json","view_paper":"https://pith.science/paper/5OZ7KHDX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.01267&json=true","fetch_graph":"https://pith.science/api/pith-number/5OZ7KHDXW5WOVF2PWXDQHFZYIS/graph.json","fetch_events":"https://pith.science/api/pith-number/5OZ7KHDXW5WOVF2PWXDQHFZYIS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS/action/storage_attestation","attest_author":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS/action/author_attestation","sign_citation":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS/action/citation_signature","submit_replication":"https://pith.science/pith/5OZ7KHDXW5WOVF2PWXDQHFZYIS/action/replication_record"}},"created_at":"2026-07-05T01:13:23.310248+00:00","updated_at":"2026-07-05T01:13:23.310248+00:00"}