{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:EXHB3GLIQI2CRJCJPV7JOGNEKZ","short_pith_number":"pith:EXHB3GLI","schema_version":"1.0","canonical_sha256":"25ce1d9968823428a4497d7e9719a4564c5079bf6df20ab702327e0b900e52de","source":{"kind":"arxiv","id":"1804.01606","version":1},"attestation_state":"computed","paper":{"title":"On the number of containments in $P$-free families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Bal\\'azs Patk\\'os, D\\'aniel Gerbner, D\\'aniel T. Nagy, M\\'at\\'e Vizer","submitted_at":"2018-04-04T21:06:53Z","abstract_excerpt":"A subfamily $\\{F_1,F_2,\\dots,F_{|P|}\\}\\subseteq \\mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\\rightarrow \\{F_1,F_2,\\dots,F_{|P|}\\}$ such that $p\\le_P q$ implies $i(p)\\subseteq i(q)$. A family $\\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this paper we establish basic results on the maximum possible number of $k$-chains in a $P$-free family $\\mathcal F\\subseteq 2^{[n]}$. We prove that if the height of $P$, $h(P) > k$, then this number is of the order $\\Theta(\\prod_{i=1}^{k+1}\\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\\ge l_2\\ge \\dots \\ge l_{k+1}"},"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":"1804.01606","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-04T21:06:53Z","cross_cats_sorted":[],"title_canon_sha256":"4b2f2687e420f1a1ca1ae82417d09de88335138a9385c6c5f04af4b45ec67706","abstract_canon_sha256":"3eac3297835dbc063f6adf6dfd1d8fd3d26b68c4714843332495c6b1a857f0c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:10.728254Z","signature_b64":"+suh2u7q94oVQ3Wxp52h47OX1jpva3sFwJrPI5d8tf/zugGddToPaySb9llCB7+qSc6e2PT/1o1WbALZZnCFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25ce1d9968823428a4497d7e9719a4564c5079bf6df20ab702327e0b900e52de","last_reissued_at":"2026-05-18T00:19:10.727603Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:10.727603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of containments in $P$-free families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Bal\\'azs Patk\\'os, D\\'aniel Gerbner, D\\'aniel T. Nagy, M\\'at\\'e Vizer","submitted_at":"2018-04-04T21:06:53Z","abstract_excerpt":"A subfamily $\\{F_1,F_2,\\dots,F_{|P|}\\}\\subseteq \\mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\\rightarrow \\{F_1,F_2,\\dots,F_{|P|}\\}$ such that $p\\le_P q$ implies $i(p)\\subseteq i(q)$. A family $\\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this paper we establish basic results on the maximum possible number of $k$-chains in a $P$-free family $\\mathcal F\\subseteq 2^{[n]}$. We prove that if the height of $P$, $h(P) > k$, then this number is of the order $\\Theta(\\prod_{i=1}^{k+1}\\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\\ge l_2\\ge \\dots \\ge l_{k+1}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01606","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":"1804.01606","created_at":"2026-05-18T00:19:10.727698+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.01606v1","created_at":"2026-05-18T00:19:10.727698+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01606","created_at":"2026-05-18T00:19:10.727698+00:00"},{"alias_kind":"pith_short_12","alias_value":"EXHB3GLIQI2C","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"EXHB3GLIQI2CRJCJ","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"EXHB3GLI","created_at":"2026-05-18T12:32:22.470017+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/EXHB3GLIQI2CRJCJPV7JOGNEKZ","json":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ.json","graph_json":"https://pith.science/api/pith-number/EXHB3GLIQI2CRJCJPV7JOGNEKZ/graph.json","events_json":"https://pith.science/api/pith-number/EXHB3GLIQI2CRJCJPV7JOGNEKZ/events.json","paper":"https://pith.science/paper/EXHB3GLI"},"agent_actions":{"view_html":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ","download_json":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ.json","view_paper":"https://pith.science/paper/EXHB3GLI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.01606&json=true","fetch_graph":"https://pith.science/api/pith-number/EXHB3GLIQI2CRJCJPV7JOGNEKZ/graph.json","fetch_events":"https://pith.science/api/pith-number/EXHB3GLIQI2CRJCJPV7JOGNEKZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ/action/storage_attestation","attest_author":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ/action/author_attestation","sign_citation":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ/action/citation_signature","submit_replication":"https://pith.science/pith/EXHB3GLIQI2CRJCJPV7JOGNEKZ/action/replication_record"}},"created_at":"2026-05-18T00:19:10.727698+00:00","updated_at":"2026-05-18T00:19:10.727698+00:00"}