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We call $\\mathcal{F} \\subset \\mathcal{P}([n])$ a \\textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \\in \\mathcal{F}$ with: $$(i) \\ \\ p|F \\setminus G|=q|G \\setminus F|, \\ \\textrm{and}$$ $$(ii) \\ f > g \\ \\textrm{for all} \\ f \\in F \\setminus G \\ \\textrm{and} \\ g \\in G \\setminus F.$$\n  Long (\\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\\sqrt{\\log n}}\\ \\frac{2^n}{\\sqrt{n}}).$$\n  We improve and generalize this result, and prove that th"},"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":"1507.02242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-08T18:08:41Z","cross_cats_sorted":[],"title_canon_sha256":"6f386e5afa8234353df112e336489555a1e972c495f87fef731d0d4163628544","abstract_canon_sha256":"c12308d042af26b08bd738df284ab37f12a5afc21c69f1f284f00f79788fd2d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:35.975129Z","signature_b64":"22jQe4og76USHOTW29j5PgeNqH6Jk9a2n9+XaYYvCAryxM7DaRAgDQAADBbfmsC9MfWXJzG+mBb3SCLIQYiXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fce1cf9e719e5fd3d2b656d718fbaa933e3eac68483c03e56902bb2550c13cc6","last_reissued_at":"2026-05-18T01:14:35.974279Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:35.974279Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on tilted Sperner families with patterns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D\\'aniel Gerbner, M\\'at\\'e Vizer","submitted_at":"2015-07-08T18:08:41Z","abstract_excerpt":"Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\\mathcal{F} \\subset \\mathcal{P}([n])$ a \\textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \\in \\mathcal{F}$ with: $$(i) \\ \\ p|F \\setminus G|=q|G \\setminus F|, \\ \\textrm{and}$$ $$(ii) \\ f > g \\ \\textrm{for all} \\ f \\in F \\setminus G \\ \\textrm{and} \\ g \\in G \\setminus F.$$\n  Long (\\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\\sqrt{\\log n}}\\ \\frac{2^n}{\\sqrt{n}}).$$\n  We improve and generalize this result, and prove that th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02242","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":"1507.02242","created_at":"2026-05-18T01:14:35.974442+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.02242v2","created_at":"2026-05-18T01:14:35.974442+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02242","created_at":"2026-05-18T01:14:35.974442+00:00"},{"alias_kind":"pith_short_12","alias_value":"7TQ47HTRTZP5","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7TQ47HTRTZP5HUVW","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7TQ47HTR","created_at":"2026-05-18T12:29:10.953037+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/7TQ47HTRTZP5HUVWK3LRR65KSM","json":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM.json","graph_json":"https://pith.science/api/pith-number/7TQ47HTRTZP5HUVWK3LRR65KSM/graph.json","events_json":"https://pith.science/api/pith-number/7TQ47HTRTZP5HUVWK3LRR65KSM/events.json","paper":"https://pith.science/paper/7TQ47HTR"},"agent_actions":{"view_html":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM","download_json":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM.json","view_paper":"https://pith.science/paper/7TQ47HTR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.02242&json=true","fetch_graph":"https://pith.science/api/pith-number/7TQ47HTRTZP5HUVWK3LRR65KSM/graph.json","fetch_events":"https://pith.science/api/pith-number/7TQ47HTRTZP5HUVWK3LRR65KSM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM/action/storage_attestation","attest_author":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM/action/author_attestation","sign_citation":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM/action/citation_signature","submit_replication":"https://pith.science/pith/7TQ47HTRTZP5HUVWK3LRR65KSM/action/replication_record"}},"created_at":"2026-05-18T01:14:35.974442+00:00","updated_at":"2026-05-18T01:14:35.974442+00:00"}