{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:MW5NPTIM45OXNC7RTZRB3LWJJ7","short_pith_number":"pith:MW5NPTIM","schema_version":"1.0","canonical_sha256":"65bad7cd0ce75d768bf19e621daec94fdc13d1d5087cfb3ef7ea12450c8d531e","source":{"kind":"arxiv","id":"2402.16730","version":1},"attestation_state":"computed","paper":{"title":"The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sumin Huang","submitted_at":"2024-02-26T16:52:25Z","abstract_excerpt":"Let $\\omega(\\mathcal{F})=\\sum_{\\{A,B\\}\\subset\\mathcal{F}}|A\\cap B|$ and $\\omega(\\mathcal{A},\\mathcal{B})=\\sum_{(A,B)\\in \\mathcal{A}\\times \\mathcal{B}}|A\\cap B|$. A family $\\mathcal{F}$ is intersecting if $F_1\\cap F_2\\neq \\emptyset$ for any $F_1,F_2\\in\\mathcal{F}$ and two family $\\mathcal{A}$ and $\\mathcal{B}$ are crossing-intersecting if $A\\cap B\\neq \\emptyset$ for any $(A,B)\\in \\mathcal{A}\\times\\mathcal{B}$. For an intersecting family $\\mathcal{F}$, Erd\\H{o}s, Ko and Rado determined the upper bound of $|\\mathcal{F}|$, consequently yielding an upper bound of $\\binom{|\\mathcal{F}|}{2}=\\sum_{\\{A"},"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":"2402.16730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-02-26T16:52:25Z","cross_cats_sorted":[],"title_canon_sha256":"7ba9f2822a62d9b96a2162c922b6b30707aec54e278be22ac2508359cdf07668","abstract_canon_sha256":"65bf731947d8c8657e05141b4a9c24383e331f3da65a746f8131b74278aaaba8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:49:25.549905Z","signature_b64":"WFvDbsz34MxcuyTEGMIyxr4Xk5UUa7Ak83CDfWFc7PwsSL4HEEh8J99XKKVKnhjHWmHgYVf40m1akqXQVcVNCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65bad7cd0ce75d768bf19e621daec94fdc13d1d5087cfb3ef7ea12450c8d531e","last_reissued_at":"2026-07-05T07:49:25.549494Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:49:25.549494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sumin Huang","submitted_at":"2024-02-26T16:52:25Z","abstract_excerpt":"Let $\\omega(\\mathcal{F})=\\sum_{\\{A,B\\}\\subset\\mathcal{F}}|A\\cap B|$ and $\\omega(\\mathcal{A},\\mathcal{B})=\\sum_{(A,B)\\in \\mathcal{A}\\times \\mathcal{B}}|A\\cap B|$. A family $\\mathcal{F}$ is intersecting if $F_1\\cap F_2\\neq \\emptyset$ for any $F_1,F_2\\in\\mathcal{F}$ and two family $\\mathcal{A}$ and $\\mathcal{B}$ are crossing-intersecting if $A\\cap B\\neq \\emptyset$ for any $(A,B)\\in \\mathcal{A}\\times\\mathcal{B}$. For an intersecting family $\\mathcal{F}$, Erd\\H{o}s, Ko and Rado determined the upper bound of $|\\mathcal{F}|$, consequently yielding an upper bound of $\\binom{|\\mathcal{F}|}{2}=\\sum_{\\{A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.16730","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2402.16730/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2402.16730","created_at":"2026-07-05T07:49:25.549553+00:00"},{"alias_kind":"arxiv_version","alias_value":"2402.16730v1","created_at":"2026-07-05T07:49:25.549553+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.16730","created_at":"2026-07-05T07:49:25.549553+00:00"},{"alias_kind":"pith_short_12","alias_value":"MW5NPTIM45OX","created_at":"2026-07-05T07:49:25.549553+00:00"},{"alias_kind":"pith_short_16","alias_value":"MW5NPTIM45OXNC7R","created_at":"2026-07-05T07:49:25.549553+00:00"},{"alias_kind":"pith_short_8","alias_value":"MW5NPTIM","created_at":"2026-07-05T07:49:25.549553+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/MW5NPTIM45OXNC7RTZRB3LWJJ7","json":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7.json","graph_json":"https://pith.science/api/pith-number/MW5NPTIM45OXNC7RTZRB3LWJJ7/graph.json","events_json":"https://pith.science/api/pith-number/MW5NPTIM45OXNC7RTZRB3LWJJ7/events.json","paper":"https://pith.science/paper/MW5NPTIM"},"agent_actions":{"view_html":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7","download_json":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7.json","view_paper":"https://pith.science/paper/MW5NPTIM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2402.16730&json=true","fetch_graph":"https://pith.science/api/pith-number/MW5NPTIM45OXNC7RTZRB3LWJJ7/graph.json","fetch_events":"https://pith.science/api/pith-number/MW5NPTIM45OXNC7RTZRB3LWJJ7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7/action/storage_attestation","attest_author":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7/action/author_attestation","sign_citation":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7/action/citation_signature","submit_replication":"https://pith.science/pith/MW5NPTIM45OXNC7RTZRB3LWJJ7/action/replication_record"}},"created_at":"2026-07-05T07:49:25.549553+00:00","updated_at":"2026-07-05T07:49:25.549553+00:00"}