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A family of sets $\\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\\binom{[n]}{k} \\subseteq \\mathcal{F} \\subseteq \\binom{[m]}{k}$ and any pair of members of $\\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large."},"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":"1304.1861","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-06T06:44:15Z","cross_cats_sorted":[],"title_canon_sha256":"0356d44048dd202e2bb966ec16f94ca164e31963e2f8e08aa847dd3f76397550","abstract_canon_sha256":"7e9620981c7140be699de4045bbc2fba34f3ce415b9d0556f1dd2eb00589adf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:44.941926Z","signature_b64":"d1M+qb/pbzlsnM/wVQ8RoyszNmstVXi30ENrtGM00E/JsV2qmL5/eUQNMHRakSYvhyfqTDHjOxjWhg+xs4Z5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f38f403b51be174fd7079eea9e1fd54c5234cc9462550def0de880d8a473a584","last_reissued_at":"2026-05-18T03:28:44.941253Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:44.941253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Common Generalization of the Theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bor-Liang Chen, Ko-Wei Lih, Kuo-Ching Huang, Wei-Tian Li","submitted_at":"2013-04-06T06:44:15Z","abstract_excerpt":"Let $m$, $n$, and $k$ be integers satisfying $0 < k \\leq n < 2k \\leq m$. A family of sets $\\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\\binom{[n]}{k} \\subseteq \\mathcal{F} \\subseteq \\binom{[m]}{k}$ and any pair of members of $\\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1861","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":"1304.1861","created_at":"2026-05-18T03:28:44.941364+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.1861v1","created_at":"2026-05-18T03:28:44.941364+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1861","created_at":"2026-05-18T03:28:44.941364+00:00"},{"alias_kind":"pith_short_12","alias_value":"6OHUAO2RXYLU","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6OHUAO2RXYLU7VYH","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6OHUAO2R","created_at":"2026-05-18T12:27:36.564083+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/6OHUAO2RXYLU7VYHT3VJ4H6VJR","json":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR.json","graph_json":"https://pith.science/api/pith-number/6OHUAO2RXYLU7VYHT3VJ4H6VJR/graph.json","events_json":"https://pith.science/api/pith-number/6OHUAO2RXYLU7VYHT3VJ4H6VJR/events.json","paper":"https://pith.science/paper/6OHUAO2R"},"agent_actions":{"view_html":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR","download_json":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR.json","view_paper":"https://pith.science/paper/6OHUAO2R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.1861&json=true","fetch_graph":"https://pith.science/api/pith-number/6OHUAO2RXYLU7VYHT3VJ4H6VJR/graph.json","fetch_events":"https://pith.science/api/pith-number/6OHUAO2RXYLU7VYHT3VJ4H6VJR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR/action/storage_attestation","attest_author":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR/action/author_attestation","sign_citation":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR/action/citation_signature","submit_replication":"https://pith.science/pith/6OHUAO2RXYLU7VYHT3VJ4H6VJR/action/replication_record"}},"created_at":"2026-05-18T03:28:44.941364+00:00","updated_at":"2026-05-18T03:28:44.941364+00:00"}