{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:LPSJUNA3SAGXJ2N5TQZRX4F4H3","short_pith_number":"pith:LPSJUNA3","schema_version":"1.0","canonical_sha256":"5be49a341b900d74e9bd9c331bf0bc3ef535eff636ad08a716d97318841dccf5","source":{"kind":"arxiv","id":"1408.0646","version":1},"attestation_state":"computed","paper":{"title":"Set families with forbidden subposets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans, Linyuan Lu","submitted_at":"2014-08-04T11:22:25Z","abstract_excerpt":"Let $F$ be a family of subsets of $\\{1,\\ldots,n\\}$. We say that $F$ is $P$-free if the inclusion order on $F$ does not contain $P$ as an induced subposet. The \\emph{Tur\\'an function} of $P$, denoted $\\pi^*(n,P)$, is the maximum size of a $P$-free family of subsets of $\\{1,\\ldots,n\\}$. We show that $\\pi^*(n,P) \\le (4r + O(\\sqrt{r}))\\binom{n}{n/2}$ if $P$ is an $r$-element poset of height at most $2$. We also show that $\\pi^*(n,S_r) = (r+O(\\sqrt{r}))\\binom{n}{n/2}$ where $S_r$ is the standard example on $2r$ elements, and that $\\pi^*(n,B_2) \\le (2.583+o(1))\\binom{n}{n/2}$, where $B_2$ is the $2$"},"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":"1408.0646","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-04T11:22:25Z","cross_cats_sorted":[],"title_canon_sha256":"c96c8d68e2101cbeec031af65d2fd4c848d6b6a9bcf7954462c59dc3fb6c9d8d","abstract_canon_sha256":"b8c7c28ebc3d6fc4d407cc1a6d056d7f64235f8d8389a7663f3a157d2f782f49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:56.585006Z","signature_b64":"kO9KsWKKR6mQ+oqPLwylQCkVxRexCWxjN/3SSy9NYFTAFq6alscByScTaFwSj8CKt0fCKENLUB5C8fIVRnMIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5be49a341b900d74e9bd9c331bf0bc3ef535eff636ad08a716d97318841dccf5","last_reissued_at":"2026-05-18T02:45:56.584158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:56.584158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Set families with forbidden subposets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans, Linyuan Lu","submitted_at":"2014-08-04T11:22:25Z","abstract_excerpt":"Let $F$ be a family of subsets of $\\{1,\\ldots,n\\}$. We say that $F$ is $P$-free if the inclusion order on $F$ does not contain $P$ as an induced subposet. The \\emph{Tur\\'an function} of $P$, denoted $\\pi^*(n,P)$, is the maximum size of a $P$-free family of subsets of $\\{1,\\ldots,n\\}$. We show that $\\pi^*(n,P) \\le (4r + O(\\sqrt{r}))\\binom{n}{n/2}$ if $P$ is an $r$-element poset of height at most $2$. We also show that $\\pi^*(n,S_r) = (r+O(\\sqrt{r}))\\binom{n}{n/2}$ where $S_r$ is the standard example on $2r$ elements, and that $\\pi^*(n,B_2) \\le (2.583+o(1))\\binom{n}{n/2}$, where $B_2$ is the $2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0646","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":"1408.0646","created_at":"2026-05-18T02:45:56.584252+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.0646v1","created_at":"2026-05-18T02:45:56.584252+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.0646","created_at":"2026-05-18T02:45:56.584252+00:00"},{"alias_kind":"pith_short_12","alias_value":"LPSJUNA3SAGX","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"LPSJUNA3SAGXJ2N5","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"LPSJUNA3","created_at":"2026-05-18T12:28:38.356838+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/LPSJUNA3SAGXJ2N5TQZRX4F4H3","json":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3.json","graph_json":"https://pith.science/api/pith-number/LPSJUNA3SAGXJ2N5TQZRX4F4H3/graph.json","events_json":"https://pith.science/api/pith-number/LPSJUNA3SAGXJ2N5TQZRX4F4H3/events.json","paper":"https://pith.science/paper/LPSJUNA3"},"agent_actions":{"view_html":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3","download_json":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3.json","view_paper":"https://pith.science/paper/LPSJUNA3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.0646&json=true","fetch_graph":"https://pith.science/api/pith-number/LPSJUNA3SAGXJ2N5TQZRX4F4H3/graph.json","fetch_events":"https://pith.science/api/pith-number/LPSJUNA3SAGXJ2N5TQZRX4F4H3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3/action/storage_attestation","attest_author":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3/action/author_attestation","sign_citation":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3/action/citation_signature","submit_replication":"https://pith.science/pith/LPSJUNA3SAGXJ2N5TQZRX4F4H3/action/replication_record"}},"created_at":"2026-05-18T02:45:56.584252+00:00","updated_at":"2026-05-18T02:45:56.584252+00:00"}