{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:DZAP55CEHZMXXP2YIJGSTV7ZAW","short_pith_number":"pith:DZAP55CE","schema_version":"1.0","canonical_sha256":"1e40fef4443e597bbf58424d29d7f9058fb30a7796091724699107b56d23e6ef","source":{"kind":"arxiv","id":"1701.04110","version":2},"attestation_state":"computed","paper":{"title":"Counting intersecting and pairs of cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2017-01-15T21:01:55Z","abstract_excerpt":"A family of subsets of $\\{1,\\ldots,n\\}$ is called {\\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\\{1,\\ldots, n\\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\\{1,\\ldots, n\\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\\ge 2k+2+2\\sqrt{k\\log k}$ and $k\\to \\infty$. Moreover, under the same assumptions we also d"},"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":"1701.04110","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-15T21:01:55Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"21cb88d8af1bbe1a2b229733487796107b2b5bc501e782845e816b44193b3508","abstract_canon_sha256":"d6c566d17728242afae57b6f63613d07de047a0d00ae5dfe67cd62c295622125"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:21.041245Z","signature_b64":"IpnM+zOha7PSnN1YG49sa0FPDn3N1FtYXb2x0aTWYfJEuAwYV8zL6vuGavaR9rq1z4QYEhQCTX30Hu/RfCmYDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e40fef4443e597bbf58424d29d7f9058fb30a7796091724699107b56d23e6ef","last_reissued_at":"2026-05-18T00:29:21.040443Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:21.040443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting intersecting and pairs of cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2017-01-15T21:01:55Z","abstract_excerpt":"A family of subsets of $\\{1,\\ldots,n\\}$ is called {\\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\\{1,\\ldots, n\\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\\{1,\\ldots, n\\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\\ge 2k+2+2\\sqrt{k\\log k}$ and $k\\to \\infty$. Moreover, under the same assumptions we also d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04110","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":"1701.04110","created_at":"2026-05-18T00:29:21.040575+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.04110v2","created_at":"2026-05-18T00:29:21.040575+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04110","created_at":"2026-05-18T00:29:21.040575+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZAP55CEHZMX","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZAP55CEHZMXXP2Y","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZAP55CE","created_at":"2026-05-18T12:31:12.930513+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/DZAP55CEHZMXXP2YIJGSTV7ZAW","json":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW.json","graph_json":"https://pith.science/api/pith-number/DZAP55CEHZMXXP2YIJGSTV7ZAW/graph.json","events_json":"https://pith.science/api/pith-number/DZAP55CEHZMXXP2YIJGSTV7ZAW/events.json","paper":"https://pith.science/paper/DZAP55CE"},"agent_actions":{"view_html":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW","download_json":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW.json","view_paper":"https://pith.science/paper/DZAP55CE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.04110&json=true","fetch_graph":"https://pith.science/api/pith-number/DZAP55CEHZMXXP2YIJGSTV7ZAW/graph.json","fetch_events":"https://pith.science/api/pith-number/DZAP55CEHZMXXP2YIJGSTV7ZAW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW/action/storage_attestation","attest_author":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW/action/author_attestation","sign_citation":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW/action/citation_signature","submit_replication":"https://pith.science/pith/DZAP55CEHZMXXP2YIJGSTV7ZAW/action/replication_record"}},"created_at":"2026-05-18T00:29:21.040575+00:00","updated_at":"2026-05-18T00:29:21.040575+00:00"}