{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:D24I7WF7SM4U2XOJR6Z7ZKYMUS","short_pith_number":"pith:D24I7WF7","schema_version":"1.0","canonical_sha256":"1eb88fd8bf93394d5dc98fb3fcab0ca4a8e993efa0ed55ceaeae71f05e4f9dc2","source":{"kind":"arxiv","id":"1509.05464","version":4},"attestation_state":"computed","paper":{"title":"The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Han, Yoshiharu Kohayakawa","submitted_at":"2015-09-17T22:35:02Z","abstract_excerpt":"The celebrated Erd\\H{o}s-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called Erd\\H{o}s-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting $k$-uniform family that is neither a subfamily of the Erd\\H{o}s-Ko-Rado family nor of the Hilton-Milner family is. For $k\\ge 4$, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting m"},"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":"1509.05464","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-17T22:35:02Z","cross_cats_sorted":[],"title_canon_sha256":"b4ed3e9487dbc7c2bdde87bf82df3fbecce04b68f629c414c0a2d9909ad04b2a","abstract_canon_sha256":"ef8aaa055399f3d87a4c4ef2bba3107736562b4a156ad517c16eba1c7f68e31a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:15.050253Z","signature_b64":"vxW6AaAaHntAI9oo5Bx6LFkcGR4H62rkaedkA1V1pOLotZbeoqmBJpuQMEJ5jDoZ9IjRX/YZvswDz3ppuvysDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1eb88fd8bf93394d5dc98fb3fcab0ca4a8e993efa0ed55ceaeae71f05e4f9dc2","last_reissued_at":"2026-05-18T01:10:15.049762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:15.049762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Han, Yoshiharu Kohayakawa","submitted_at":"2015-09-17T22:35:02Z","abstract_excerpt":"The celebrated Erd\\H{o}s-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called Erd\\H{o}s-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting $k$-uniform family that is neither a subfamily of the Erd\\H{o}s-Ko-Rado family nor of the Hilton-Milner family is. For $k\\ge 4$, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05464","kind":"arxiv","version":4},"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":"1509.05464","created_at":"2026-05-18T01:10:15.049848+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05464v4","created_at":"2026-05-18T01:10:15.049848+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05464","created_at":"2026-05-18T01:10:15.049848+00:00"},{"alias_kind":"pith_short_12","alias_value":"D24I7WF7SM4U","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"D24I7WF7SM4U2XOJ","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"D24I7WF7","created_at":"2026-05-18T12:29:17.054201+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/D24I7WF7SM4U2XOJR6Z7ZKYMUS","json":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS.json","graph_json":"https://pith.science/api/pith-number/D24I7WF7SM4U2XOJR6Z7ZKYMUS/graph.json","events_json":"https://pith.science/api/pith-number/D24I7WF7SM4U2XOJR6Z7ZKYMUS/events.json","paper":"https://pith.science/paper/D24I7WF7"},"agent_actions":{"view_html":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS","download_json":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS.json","view_paper":"https://pith.science/paper/D24I7WF7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05464&json=true","fetch_graph":"https://pith.science/api/pith-number/D24I7WF7SM4U2XOJR6Z7ZKYMUS/graph.json","fetch_events":"https://pith.science/api/pith-number/D24I7WF7SM4U2XOJR6Z7ZKYMUS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS/action/storage_attestation","attest_author":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS/action/author_attestation","sign_citation":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS/action/citation_signature","submit_replication":"https://pith.science/pith/D24I7WF7SM4U2XOJR6Z7ZKYMUS/action/replication_record"}},"created_at":"2026-05-18T01:10:15.049848+00:00","updated_at":"2026-05-18T01:10:15.049848+00:00"}