{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:AIFLI4H5Q3TMCXBETPJ2PSXOGI","short_pith_number":"pith:AIFLI4H5","schema_version":"1.0","canonical_sha256":"020ab470fd86e6c15c249bd3a7caee323a6801f5fa186b728040e286978d1a4f","source":{"kind":"arxiv","id":"1708.02436","version":1},"attestation_state":"computed","paper":{"title":"Subsets of posets minimising the number of chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wojciech Samotij","submitted_at":"2017-08-08T10:09:42Z","abstract_excerpt":"A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\\H{o}s characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \\subset \\dotsc \\subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to"},"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":"1708.02436","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-08T10:09:42Z","cross_cats_sorted":[],"title_canon_sha256":"1ece5fb9da49a0e095b16c53de269514bb7d3f24419ba9aa20bd632b3098b467","abstract_canon_sha256":"d99bbc55eff25cc2fc74f0f36e68da6e4de37a221a21bab1171a7f44f9cfdbd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:24.277581Z","signature_b64":"Ka5lZUxn1FUYQcWnrEEEKgcc029Nidg0M4xfLJs6OvJkHR3J0dXlOj3oUIDvAVekhRZUEfNJsb0QdzpxwEb5CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"020ab470fd86e6c15c249bd3a7caee323a6801f5fa186b728040e286978d1a4f","last_reissued_at":"2026-05-18T00:38:24.276824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:24.276824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subsets of posets minimising the number of chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wojciech Samotij","submitted_at":"2017-08-08T10:09:42Z","abstract_excerpt":"A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\\H{o}s characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \\subset \\dotsc \\subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02436","kind":"arxiv","version":1},"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":"1708.02436","created_at":"2026-05-18T00:38:24.276947+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.02436v1","created_at":"2026-05-18T00:38:24.276947+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02436","created_at":"2026-05-18T00:38:24.276947+00:00"},{"alias_kind":"pith_short_12","alias_value":"AIFLI4H5Q3TM","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AIFLI4H5Q3TMCXBE","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AIFLI4H5","created_at":"2026-05-18T12:31:05.417338+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/AIFLI4H5Q3TMCXBETPJ2PSXOGI","json":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI.json","graph_json":"https://pith.science/api/pith-number/AIFLI4H5Q3TMCXBETPJ2PSXOGI/graph.json","events_json":"https://pith.science/api/pith-number/AIFLI4H5Q3TMCXBETPJ2PSXOGI/events.json","paper":"https://pith.science/paper/AIFLI4H5"},"agent_actions":{"view_html":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI","download_json":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI.json","view_paper":"https://pith.science/paper/AIFLI4H5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.02436&json=true","fetch_graph":"https://pith.science/api/pith-number/AIFLI4H5Q3TMCXBETPJ2PSXOGI/graph.json","fetch_events":"https://pith.science/api/pith-number/AIFLI4H5Q3TMCXBETPJ2PSXOGI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI/action/storage_attestation","attest_author":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI/action/author_attestation","sign_citation":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI/action/citation_signature","submit_replication":"https://pith.science/pith/AIFLI4H5Q3TMCXBETPJ2PSXOGI/action/replication_record"}},"created_at":"2026-05-18T00:38:24.276947+00:00","updated_at":"2026-05-18T00:38:24.276947+00:00"}