{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:NUPWQ3LB6A7Q2AHYXV2LXSP4W6","short_pith_number":"pith:NUPWQ3LB","schema_version":"1.0","canonical_sha256":"6d1f686d61f03f0d00f8bd74bbc9fcb7b282ec12e0e9d96bf9ad57d2a9ad5902","source":{"kind":"arxiv","id":"0911.4204","version":2},"attestation_state":"computed","paper":{"title":"Maximal independent sets and separating covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vincent Vatter","submitted_at":"2009-11-21T21:40:18Z","abstract_excerpt":"In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns out to be the inverse to the answer to a much simpler question: what is the largest integer which is the product of positive integers with sum n? We give a combinatorial explanation for this relationship, via Moon and Moser's answer to a question of Erdos: how many maximal independent sets can a graph on n vertices have? We conclude by showing how Moon and Moser's solution also sheds light on a problem of Mahler and Popken's about the comple"},"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":"0911.4204","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T21:40:18Z","cross_cats_sorted":[],"title_canon_sha256":"536d3b18615ecb7105872cac0a4e35309478273893a3bcc35ae4237231db86e0","abstract_canon_sha256":"8dfc86ee131a3d903804b138f2afb74de6bcdfcd23e3a7942b3b63004cc80e7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:44.857031Z","signature_b64":"VByF6tOynW8EI3Ij7JXFCJL1Lmd9FptBNXuIscRc9xv7RpYEMyCY/WA3RJBkSVdqL3pN7ImIZv6LFulUU8O2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d1f686d61f03f0d00f8bd74bbc9fcb7b282ec12e0e9d96bf9ad57d2a9ad5902","last_reissued_at":"2026-05-18T04:41:44.856665Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:44.856665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximal independent sets and separating covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vincent Vatter","submitted_at":"2009-11-21T21:40:18Z","abstract_excerpt":"In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns out to be the inverse to the answer to a much simpler question: what is the largest integer which is the product of positive integers with sum n? We give a combinatorial explanation for this relationship, via Moon and Moser's answer to a question of Erdos: how many maximal independent sets can a graph on n vertices have? We conclude by showing how Moon and Moser's solution also sheds light on a problem of Mahler and Popken's about the comple"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4204","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":"0911.4204","created_at":"2026-05-18T04:41:44.856725+00:00"},{"alias_kind":"arxiv_version","alias_value":"0911.4204v2","created_at":"2026-05-18T04:41:44.856725+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4204","created_at":"2026-05-18T04:41:44.856725+00:00"},{"alias_kind":"pith_short_12","alias_value":"NUPWQ3LB6A7Q","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"NUPWQ3LB6A7Q2AHY","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"NUPWQ3LB","created_at":"2026-05-18T12:26:00.592388+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/NUPWQ3LB6A7Q2AHYXV2LXSP4W6","json":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6.json","graph_json":"https://pith.science/api/pith-number/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/graph.json","events_json":"https://pith.science/api/pith-number/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/events.json","paper":"https://pith.science/paper/NUPWQ3LB"},"agent_actions":{"view_html":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6","download_json":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6.json","view_paper":"https://pith.science/paper/NUPWQ3LB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0911.4204&json=true","fetch_graph":"https://pith.science/api/pith-number/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/graph.json","fetch_events":"https://pith.science/api/pith-number/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/action/storage_attestation","attest_author":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/action/author_attestation","sign_citation":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/action/citation_signature","submit_replication":"https://pith.science/pith/NUPWQ3LB6A7Q2AHYXV2LXSP4W6/action/replication_record"}},"created_at":"2026-05-18T04:41:44.856725+00:00","updated_at":"2026-05-18T04:41:44.856725+00:00"}