{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:D24I7WF7SM4U2XOJR6Z7ZKYMUS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ef8aaa055399f3d87a4c4ef2bba3107736562b4a156ad517c16eba1c7f68e31a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-17T22:35:02Z","title_canon_sha256":"b4ed3e9487dbc7c2bdde87bf82df3fbecce04b68f629c414c0a2d9909ad04b2a"},"schema_version":"1.0","source":{"id":"1509.05464","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05464","created_at":"2026-05-18T01:10:15Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05464v4","created_at":"2026-05-18T01:10:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05464","created_at":"2026-05-18T01:10:15Z"},{"alias_kind":"pith_short_12","alias_value":"D24I7WF7SM4U","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"D24I7WF7SM4U2XOJ","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"D24I7WF7","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:ab306407c85b3bed889d0c2eb4ea924dcb483979aabe7ce23525eabf032783e2","target":"graph","created_at":"2026-05-18T01:10:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Jie Han, Yoshiharu Kohayakawa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-17T22:35:02Z","title":"The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05464","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2c6f51a79060c2912eceac9494a01ed102575d3546d76907c8ad579aedf9fb58","target":"record","created_at":"2026-05-18T01:10:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ef8aaa055399f3d87a4c4ef2bba3107736562b4a156ad517c16eba1c7f68e31a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-17T22:35:02Z","title_canon_sha256":"b4ed3e9487dbc7c2bdde87bf82df3fbecce04b68f629c414c0a2d9909ad04b2a"},"schema_version":"1.0","source":{"id":"1509.05464","kind":"arxiv","version":4}},"canonical_sha256":"1eb88fd8bf93394d5dc98fb3fcab0ca4a8e993efa0ed55ceaeae71f05e4f9dc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1eb88fd8bf93394d5dc98fb3fcab0ca4a8e993efa0ed55ceaeae71f05e4f9dc2","first_computed_at":"2026-05-18T01:10:15.049762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:15.049762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vxW6AaAaHntAI9oo5Bx6LFkcGR4H62rkaedkA1V1pOLotZbeoqmBJpuQMEJ5jDoZ9IjRX/YZvswDz3ppuvysDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:15.050253Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05464","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c6f51a79060c2912eceac9494a01ed102575d3546d76907c8ad579aedf9fb58","sha256:ab306407c85b3bed889d0c2eb4ea924dcb483979aabe7ce23525eabf032783e2"],"state_sha256":"6512683dee55228c0bcab8b1ecdea696f547a676273cfd4f74d8f2e3a8deb848"}