{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2QPQ4CJLHREKMVRN3LXVXPL7YQ","short_pith_number":"pith:2QPQ4CJL","schema_version":"1.0","canonical_sha256":"d41f0e092b3c48a6562ddaef5bbd7fc435b32dff370d62f85bd7b617471d0f18","source":{"kind":"arxiv","id":"1603.00936","version":2},"attestation_state":"computed","paper":{"title":"A size-sensitive inequality for 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":"2016-03-03T00:17:28Z","abstract_excerpt":"Two families $\\mathcal A$ and $\\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\\cap B\\ne\\emptyset$ for all $A\\in \\mathcal A, B\\in \\mathcal B $. Strengthening the classical Erd\\H os-Ko-Rado theorem, Pyber proved that $|\\mathcal A||\\mathcal B|\\le {n-1\\choose k-1}^2$ holds for $n\\ge 2k$. In the present paper we sharpen this inequality. We prove that assuming $|\\mathcal B|\\ge {n-1\\choose k-1}+{n-i\\choose k-i+1}$ for some $3\\le i\\le k+1$ the stronger inequality $$|\\mathcal A||\\mathcal B|\\le \\Bigl({n-1\\choose k-1}+{n-i\\choose k-i+1}\\Bigr)\\Bigl({n-1\\choose k-1}-{n-i\\choose"},"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":"1603.00936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-03T00:17:28Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"1575273fffee9b032287f59f96054a127ab6f6517411605c10fae4bf500312d8","abstract_canon_sha256":"bc6ba095fafc9b94daa26561525778c7476c035611092d9b3a57b4727b9c74a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:15.688432Z","signature_b64":"kqC7OuOwRAPsXOeMYOkWJQLwBgLy0Pqd8OB48rqaAbNIRE5KweAJYc7XLLq2NXYkmT1N+/WmPnvL+1eiLUCyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d41f0e092b3c48a6562ddaef5bbd7fc435b32dff370d62f85bd7b617471d0f18","last_reissued_at":"2026-05-18T00:29:15.687862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:15.687862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A size-sensitive inequality for 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":"2016-03-03T00:17:28Z","abstract_excerpt":"Two families $\\mathcal A$ and $\\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\\cap B\\ne\\emptyset$ for all $A\\in \\mathcal A, B\\in \\mathcal B $. Strengthening the classical Erd\\H os-Ko-Rado theorem, Pyber proved that $|\\mathcal A||\\mathcal B|\\le {n-1\\choose k-1}^2$ holds for $n\\ge 2k$. In the present paper we sharpen this inequality. We prove that assuming $|\\mathcal B|\\ge {n-1\\choose k-1}+{n-i\\choose k-i+1}$ for some $3\\le i\\le k+1$ the stronger inequality $$|\\mathcal A||\\mathcal B|\\le \\Bigl({n-1\\choose k-1}+{n-i\\choose k-i+1}\\Bigr)\\Bigl({n-1\\choose k-1}-{n-i\\choose"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00936","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":"1603.00936","created_at":"2026-05-18T00:29:15.687945+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.00936v2","created_at":"2026-05-18T00:29:15.687945+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00936","created_at":"2026-05-18T00:29:15.687945+00:00"},{"alias_kind":"pith_short_12","alias_value":"2QPQ4CJLHREK","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2QPQ4CJLHREKMVRN","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2QPQ4CJL","created_at":"2026-05-18T12:29:55.572404+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/2QPQ4CJLHREKMVRN3LXVXPL7YQ","json":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ.json","graph_json":"https://pith.science/api/pith-number/2QPQ4CJLHREKMVRN3LXVXPL7YQ/graph.json","events_json":"https://pith.science/api/pith-number/2QPQ4CJLHREKMVRN3LXVXPL7YQ/events.json","paper":"https://pith.science/paper/2QPQ4CJL"},"agent_actions":{"view_html":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ","download_json":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ.json","view_paper":"https://pith.science/paper/2QPQ4CJL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.00936&json=true","fetch_graph":"https://pith.science/api/pith-number/2QPQ4CJLHREKMVRN3LXVXPL7YQ/graph.json","fetch_events":"https://pith.science/api/pith-number/2QPQ4CJLHREKMVRN3LXVXPL7YQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ/action/storage_attestation","attest_author":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ/action/author_attestation","sign_citation":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ/action/citation_signature","submit_replication":"https://pith.science/pith/2QPQ4CJLHREKMVRN3LXVXPL7YQ/action/replication_record"}},"created_at":"2026-05-18T00:29:15.687945+00:00","updated_at":"2026-05-18T00:29:15.687945+00:00"}