{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:NMN44K2SQAFTBRR3QTZJYZLHVZ","short_pith_number":"pith:NMN44K2S","schema_version":"1.0","canonical_sha256":"6b1bce2b52800b30c63b84f29c6567ae434683abb24d1395bc37cb487f4114e6","source":{"kind":"arxiv","id":"1609.04714","version":3},"attestation_state":"computed","paper":{"title":"New injective proofs of the Erd\\H{o}s--Ko--Rado and Hilton--Milner theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Glenn Hurlbert, Vikram Kamat","submitted_at":"2016-09-15T16:07:22Z","abstract_excerpt":"A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd\\H{o}s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality of F is at most the cardinality of the family of all r-subsets of [n] containing a fixed element. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This characterization was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Er"},"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":"1609.04714","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T16:07:22Z","cross_cats_sorted":[],"title_canon_sha256":"30d5d44e564e5274959413a4e17edd70dd3f93ecbd6af3040603025e0aa2cbd8","abstract_canon_sha256":"843600607a6a7efc1d0f2aae1cac1ce25a8a6ffb9d85fd420d8575e564c25089"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:44.326138Z","signature_b64":"ZvtKd7gsz+0AgvuEXOaSwwo+AJyYFsuv05GZdluGiEoa/q7UcHuoOpGEZjKi7k6mVxUA73c3waKil0ccSofACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b1bce2b52800b30c63b84f29c6567ae434683abb24d1395bc37cb487f4114e6","last_reissued_at":"2026-05-18T00:32:44.325512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:44.325512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New injective proofs of the Erd\\H{o}s--Ko--Rado and Hilton--Milner theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Glenn Hurlbert, Vikram Kamat","submitted_at":"2016-09-15T16:07:22Z","abstract_excerpt":"A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd\\H{o}s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality of F is at most the cardinality of the family of all r-subsets of [n] containing a fixed element. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This characterization was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Er"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04714","kind":"arxiv","version":3},"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":"1609.04714","created_at":"2026-05-18T00:32:44.325600+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04714v3","created_at":"2026-05-18T00:32:44.325600+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04714","created_at":"2026-05-18T00:32:44.325600+00:00"},{"alias_kind":"pith_short_12","alias_value":"NMN44K2SQAFT","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"NMN44K2SQAFTBRR3","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"NMN44K2S","created_at":"2026-05-18T12:30:32.724797+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/NMN44K2SQAFTBRR3QTZJYZLHVZ","json":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ.json","graph_json":"https://pith.science/api/pith-number/NMN44K2SQAFTBRR3QTZJYZLHVZ/graph.json","events_json":"https://pith.science/api/pith-number/NMN44K2SQAFTBRR3QTZJYZLHVZ/events.json","paper":"https://pith.science/paper/NMN44K2S"},"agent_actions":{"view_html":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ","download_json":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ.json","view_paper":"https://pith.science/paper/NMN44K2S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04714&json=true","fetch_graph":"https://pith.science/api/pith-number/NMN44K2SQAFTBRR3QTZJYZLHVZ/graph.json","fetch_events":"https://pith.science/api/pith-number/NMN44K2SQAFTBRR3QTZJYZLHVZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ/action/storage_attestation","attest_author":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ/action/author_attestation","sign_citation":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ/action/citation_signature","submit_replication":"https://pith.science/pith/NMN44K2SQAFTBRR3QTZJYZLHVZ/action/replication_record"}},"created_at":"2026-05-18T00:32:44.325600+00:00","updated_at":"2026-05-18T00:32:44.325600+00:00"}