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For a family ${\\cal F}$ of sets, let $c(P,{\\cal F})$ denote the number of copies of $P$ in ${\\cal F}$, and we say that ${\\cal F}$ is $P$-free if $c(P,{\\cal F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families ${\\cal F} \\subseteq 2^{[n]}$, i.e. $\\max\\{c(Q,{\\cal F"},"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":"1701.05030","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-18T12:21:07Z","cross_cats_sorted":[],"title_canon_sha256":"e846d3054c428c3af749a4f5a8faa1bbf916f9e2c9732176c9fe87512dc2f986","abstract_canon_sha256":"d0244e4dff657ef966475619bf914d9d2cd8dd9126ccf9f142fc627490befd31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:30.502452Z","signature_b64":"ile3gfNpf1mjgEVpPtdJ71lUJTRcC74yN6hDrlWwpBt3sjCI0LNjmEBnTkaTLyxmDleTTs4z4rX5B5crC/CDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92fb27de5d06ec1bcad0dd5d4595cf3a0ee9f12d2cdee29ce28af596eea1c081","last_reissued_at":"2026-05-18T00:38:30.501593Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:30.501593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized forbidden subposet problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Keszegh, Balazs Patkos, Daniel Gerbner","submitted_at":"2017-01-18T12:21:07Z","abstract_excerpt":"A subfamily $\\{F_1,F_2,\\dots,F_{|P|}\\}\\subseteq {\\cal F}$ of sets is a copy of a poset $P$ in ${\\cal F}$ if there exists a bijection $\\phi:P\\rightarrow \\{F_1,F_2,\\dots,F_{|P|}\\}$ such that whenever $x \\le_P x'$ holds, then so does $\\phi(x)\\subseteq \\phi(x')$. For a family ${\\cal F}$ of sets, let $c(P,{\\cal F})$ denote the number of copies of $P$ in ${\\cal F}$, and we say that ${\\cal F}$ is $P$-free if $c(P,{\\cal F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families ${\\cal F} \\subseteq 2^{[n]}$, i.e. $\\max\\{c(Q,{\\cal F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05030","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":"1701.05030","created_at":"2026-05-18T00:38:30.501746+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05030v2","created_at":"2026-05-18T00:38:30.501746+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05030","created_at":"2026-05-18T00:38:30.501746+00:00"},{"alias_kind":"pith_short_12","alias_value":"SL5SPXS5A3WB","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SL5SPXS5A3WBXSWQ","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SL5SPXS5","created_at":"2026-05-18T12:31:43.269735+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/SL5SPXS5A3WBXSWQ3VOULFOPHI","json":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI.json","graph_json":"https://pith.science/api/pith-number/SL5SPXS5A3WBXSWQ3VOULFOPHI/graph.json","events_json":"https://pith.science/api/pith-number/SL5SPXS5A3WBXSWQ3VOULFOPHI/events.json","paper":"https://pith.science/paper/SL5SPXS5"},"agent_actions":{"view_html":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI","download_json":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI.json","view_paper":"https://pith.science/paper/SL5SPXS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05030&json=true","fetch_graph":"https://pith.science/api/pith-number/SL5SPXS5A3WBXSWQ3VOULFOPHI/graph.json","fetch_events":"https://pith.science/api/pith-number/SL5SPXS5A3WBXSWQ3VOULFOPHI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI/action/storage_attestation","attest_author":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI/action/author_attestation","sign_citation":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI/action/citation_signature","submit_replication":"https://pith.science/pith/SL5SPXS5A3WBXSWQ3VOULFOPHI/action/replication_record"}},"created_at":"2026-05-18T00:38:30.501746+00:00","updated_at":"2026-05-18T00:38:30.501746+00:00"}