{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:RAO6T4USEREYTOHVVEV47ALLTP","short_pith_number":"pith:RAO6T4US","schema_version":"1.0","canonical_sha256":"881de9f292244989b8f5a92bcf816b9bff2879711329a6905a644265b95d76bb","source":{"kind":"arxiv","id":"1406.1887","version":2},"attestation_state":"computed","paper":{"title":"Supersaturation and stability for forbidden subposet problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Patkos","submitted_at":"2014-06-07T11:32:29Z","abstract_excerpt":"We address a supersaturation problem in the context of forbidden subposets. A family $\\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \\rightarrow \\mathcal{F}$ such that $p \\le_P q$ implies $i(p) \\subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \\le c,d$ is called butterfly. The maximum size of a family $\\mathcal{F} \\subseteq 2^{[n]}$ that does not contain a butterfly is $\\Sigma(n,2)=\\binom{n}{\\lfloor n/2 \\rfloor}+\\binom{n}{\\lfloor n/2 \\rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\\mathcal{F} \\subseteq 2^{[n]}$ conta"},"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":"1406.1887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-07T11:32:29Z","cross_cats_sorted":[],"title_canon_sha256":"8cb6e9553632534e4a1851ede1011bb6af01109e594c377447cbda6bd70b0616","abstract_canon_sha256":"0d5612c10922fd52ec5089dad4e44f22dfbbb6a2f1f7dc9e5159499ba53d43d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:21.532199Z","signature_b64":"LFj6bpHIGeQQUWSssWD/ES2WKUIuN35ylfGRT9d1KqNNRCwaRnaWa2+2rtld11LCRhGQB9mnwQK9A82EulWlAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"881de9f292244989b8f5a92bcf816b9bff2879711329a6905a644265b95d76bb","last_reissued_at":"2026-05-18T01:37:21.531671Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:21.531671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Supersaturation and stability for forbidden subposet problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Patkos","submitted_at":"2014-06-07T11:32:29Z","abstract_excerpt":"We address a supersaturation problem in the context of forbidden subposets. A family $\\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \\rightarrow \\mathcal{F}$ such that $p \\le_P q$ implies $i(p) \\subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \\le c,d$ is called butterfly. The maximum size of a family $\\mathcal{F} \\subseteq 2^{[n]}$ that does not contain a butterfly is $\\Sigma(n,2)=\\binom{n}{\\lfloor n/2 \\rfloor}+\\binom{n}{\\lfloor n/2 \\rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\\mathcal{F} \\subseteq 2^{[n]}$ conta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1887","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":"1406.1887","created_at":"2026-05-18T01:37:21.531756+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.1887v2","created_at":"2026-05-18T01:37:21.531756+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.1887","created_at":"2026-05-18T01:37:21.531756+00:00"},{"alias_kind":"pith_short_12","alias_value":"RAO6T4USEREY","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"RAO6T4USEREYTOHV","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"RAO6T4US","created_at":"2026-05-18T12:28:46.137349+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/RAO6T4USEREYTOHVVEV47ALLTP","json":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP.json","graph_json":"https://pith.science/api/pith-number/RAO6T4USEREYTOHVVEV47ALLTP/graph.json","events_json":"https://pith.science/api/pith-number/RAO6T4USEREYTOHVVEV47ALLTP/events.json","paper":"https://pith.science/paper/RAO6T4US"},"agent_actions":{"view_html":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP","download_json":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP.json","view_paper":"https://pith.science/paper/RAO6T4US","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.1887&json=true","fetch_graph":"https://pith.science/api/pith-number/RAO6T4USEREYTOHVVEV47ALLTP/graph.json","fetch_events":"https://pith.science/api/pith-number/RAO6T4USEREYTOHVVEV47ALLTP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP/action/storage_attestation","attest_author":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP/action/author_attestation","sign_citation":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP/action/citation_signature","submit_replication":"https://pith.science/pith/RAO6T4USEREYTOHVVEV47ALLTP/action/replication_record"}},"created_at":"2026-05-18T01:37:21.531756+00:00","updated_at":"2026-05-18T01:37:21.531756+00:00"}