{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:ZZCSLL2V6CUXPNUD2VCSDHSHX3","short_pith_number":"pith:ZZCSLL2V","schema_version":"1.0","canonical_sha256":"ce4525af55f0a977b683d545219e47becde985bd3e13884f625f63e9c74f7bb4","source":{"kind":"arxiv","id":"1804.11269","version":1},"attestation_state":"computed","paper":{"title":"Two extremal problems on intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hao Huang","submitted_at":"2018-04-30T15:12:33Z","abstract_excerpt":"In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$, they must satisfy $\\min\\{|\\mathcal{A}|, |\\mathcal{B}|\\} \\le \\frac{1}{2} \\binom{n-1}{k-1}$? We give an affirmative answer for $n \\ge 2k^2$, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for $n \\ge 3k$, 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":"1804.11269","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-30T15:12:33Z","cross_cats_sorted":[],"title_canon_sha256":"31da352b3c49bdf1da75df9ac6216cae81b0f39afa9972d7e55f3b41b55a7fb5","abstract_canon_sha256":"ddf51788afeae67cb3bab08955f08cf20e3a0fa878bff3cba7fb8f56b6f151f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:14.074187Z","signature_b64":"7L5REnOkb6Y8XvLWAeCdpfQMVXxFIMwZB10ePWHYp/CMPOqk/DMOrWZ0HMoBOd6pfP9OGAX/kI3PkRUWttL9Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce4525af55f0a977b683d545219e47becde985bd3e13884f625f63e9c74f7bb4","last_reissued_at":"2026-05-18T00:17:14.073594Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:14.073594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two extremal problems on intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hao Huang","submitted_at":"2018-04-30T15:12:33Z","abstract_excerpt":"In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$, they must satisfy $\\min\\{|\\mathcal{A}|, |\\mathcal{B}|\\} \\le \\frac{1}{2} \\binom{n-1}{k-1}$? We give an affirmative answer for $n \\ge 2k^2$, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for $n \\ge 3k$, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.11269","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":"1804.11269","created_at":"2026-05-18T00:17:14.073652+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.11269v1","created_at":"2026-05-18T00:17:14.073652+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.11269","created_at":"2026-05-18T00:17:14.073652+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZZCSLL2V6CUX","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZZCSLL2V6CUXPNUD","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZZCSLL2V","created_at":"2026-05-18T12:33:07.085635+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/ZZCSLL2V6CUXPNUD2VCSDHSHX3","json":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3.json","graph_json":"https://pith.science/api/pith-number/ZZCSLL2V6CUXPNUD2VCSDHSHX3/graph.json","events_json":"https://pith.science/api/pith-number/ZZCSLL2V6CUXPNUD2VCSDHSHX3/events.json","paper":"https://pith.science/paper/ZZCSLL2V"},"agent_actions":{"view_html":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3","download_json":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3.json","view_paper":"https://pith.science/paper/ZZCSLL2V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.11269&json=true","fetch_graph":"https://pith.science/api/pith-number/ZZCSLL2V6CUXPNUD2VCSDHSHX3/graph.json","fetch_events":"https://pith.science/api/pith-number/ZZCSLL2V6CUXPNUD2VCSDHSHX3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3/action/storage_attestation","attest_author":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3/action/author_attestation","sign_citation":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3/action/citation_signature","submit_replication":"https://pith.science/pith/ZZCSLL2V6CUXPNUD2VCSDHSHX3/action/replication_record"}},"created_at":"2026-05-18T00:17:14.073652+00:00","updated_at":"2026-05-18T00:17:14.073652+00:00"}