{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KWIRKMD35KU7F57CLAJJCXRTAO","short_pith_number":"pith:KWIRKMD3","schema_version":"1.0","canonical_sha256":"559115307beaa9f2f7e25812915e3303a9c8803f8993e780fbdc92e43336fd2d","source":{"kind":"arxiv","id":"1206.3007","version":3},"attestation_state":"computed","paper":{"title":"Maximal antichains of minimum size","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ian T. Roberts, Thomas Kalinowski, Uwe Leck","submitted_at":"2012-06-14T05:13:34Z","abstract_excerpt":"Let $n\\geqslant 4$ be a natural number, and let $K$ be a set $K\\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\\mathcal{A}$ of subsets of $[n]$ such that $\\mathcal{A}$ contains only sets whose size is in $K$, and $A\\not\\subseteq B$ for all ${A,B}\\subseteq\\mathcal{A}$, i.e. $\\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our constru"},"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":"1206.3007","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-14T05:13:34Z","cross_cats_sorted":[],"title_canon_sha256":"d519618c2b323e17ea01b7336b0d2e16b54548bcdc4711ab4e22feb566008a37","abstract_canon_sha256":"00565430e2eea52c2f0c76615964e696259a28f6a43dc2c09dccd29661847392"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:30.579493Z","signature_b64":"csoHfdMv+VmdOa/jMokiExnVGPqM8l2YF2Y/bKJTAvmF6uaBL2vDNYqxOWBRAP3na/+abSosTjOYLwF58r0WAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"559115307beaa9f2f7e25812915e3303a9c8803f8993e780fbdc92e43336fd2d","last_reissued_at":"2026-05-18T03:28:30.578833Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:30.578833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximal antichains of minimum size","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ian T. Roberts, Thomas Kalinowski, Uwe Leck","submitted_at":"2012-06-14T05:13:34Z","abstract_excerpt":"Let $n\\geqslant 4$ be a natural number, and let $K$ be a set $K\\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\\mathcal{A}$ of subsets of $[n]$ such that $\\mathcal{A}$ contains only sets whose size is in $K$, and $A\\not\\subseteq B$ for all ${A,B}\\subseteq\\mathcal{A}$, i.e. $\\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our constru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3007","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":"1206.3007","created_at":"2026-05-18T03:28:30.578930+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.3007v3","created_at":"2026-05-18T03:28:30.578930+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.3007","created_at":"2026-05-18T03:28:30.578930+00:00"},{"alias_kind":"pith_short_12","alias_value":"KWIRKMD35KU7","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KWIRKMD35KU7F57C","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KWIRKMD3","created_at":"2026-05-18T12:27:11.947152+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/KWIRKMD35KU7F57CLAJJCXRTAO","json":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO.json","graph_json":"https://pith.science/api/pith-number/KWIRKMD35KU7F57CLAJJCXRTAO/graph.json","events_json":"https://pith.science/api/pith-number/KWIRKMD35KU7F57CLAJJCXRTAO/events.json","paper":"https://pith.science/paper/KWIRKMD3"},"agent_actions":{"view_html":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO","download_json":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO.json","view_paper":"https://pith.science/paper/KWIRKMD3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.3007&json=true","fetch_graph":"https://pith.science/api/pith-number/KWIRKMD35KU7F57CLAJJCXRTAO/graph.json","fetch_events":"https://pith.science/api/pith-number/KWIRKMD35KU7F57CLAJJCXRTAO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO/action/storage_attestation","attest_author":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO/action/author_attestation","sign_citation":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO/action/citation_signature","submit_replication":"https://pith.science/pith/KWIRKMD35KU7F57CLAJJCXRTAO/action/replication_record"}},"created_at":"2026-05-18T03:28:30.578930+00:00","updated_at":"2026-05-18T03:28:30.578930+00:00"}