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For a constant $C\\geq 1$ the $C$-diversity, $\\gamma_C(\\mathcal{F})$ is defined as $|\\mathcal{F}|-C\\Delta(\\mathcal{F})$. Define $\\mathcal{F}_{123} =\\left\\{F\\in \\binom{X}{k}\\colon |F\\cap \\{1,2,3\\}|=2\\right\\}$. It has $C$-diversity $(3-2C)\\binom{n-3}{k-2}$. The main result shows that for $1< C<\\frac{3}{2}$ and $n\\geq \\frac{42}{3-2C}k"},"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":"2308.14028","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-08-27T07:41:29Z","cross_cats_sorted":[],"title_canon_sha256":"0401e5cdb1fb0930691657419a6332b115320255a4507b5b41c8d5faa22cc2af","abstract_canon_sha256":"f1d2ba56f78d6511224b12fb812c0766df44d4903a31ceec645df45ecc4cfad1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:46:42.696260Z","signature_b64":"DTeGAwkTImCEv8DkYc6Te3gE5Xjs0AKrRKoWUwToduUlXUc7infH5Oem8cGVwsObbkGXj3f+1YtSO7MjObd0BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"093e089d44fab780790f16b0d896b2995f3997abf28ed7387823dc778893a6ef","last_reissued_at":"2026-07-05T09:46:42.695676Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:46:42.695676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $C$-diversity of intersecting hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Peter Frankl","submitted_at":"2023-08-27T07:41:29Z","abstract_excerpt":"Let $\\mathcal{F}\\subset \\binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\\mathcal{F}$ is intersecting, i.e., $F\\cap F'\\neq \\emptyset$ for all $F,F'\\in \\mathcal{F}$. Let $\\Delta(\\mathcal{F})$ be the maximum degree of $\\mathcal{F}$. For a constant $C\\geq 1$ the $C$-diversity, $\\gamma_C(\\mathcal{F})$ is defined as $|\\mathcal{F}|-C\\Delta(\\mathcal{F})$. Define $\\mathcal{F}_{123} =\\left\\{F\\in \\binom{X}{k}\\colon |F\\cap \\{1,2,3\\}|=2\\right\\}$. It has $C$-diversity $(3-2C)\\binom{n-3}{k-2}$. The main result shows that for $1< C<\\frac{3}{2}$ and $n\\geq \\frac{42}{3-2C}k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.14028","kind":"arxiv","version":2},"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/2308.14028/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":"2308.14028","created_at":"2026-07-05T09:46:42.695738+00:00"},{"alias_kind":"arxiv_version","alias_value":"2308.14028v2","created_at":"2026-07-05T09:46:42.695738+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.14028","created_at":"2026-07-05T09:46:42.695738+00:00"},{"alias_kind":"pith_short_12","alias_value":"BE7ARHKE7K3Y","created_at":"2026-07-05T09:46:42.695738+00:00"},{"alias_kind":"pith_short_16","alias_value":"BE7ARHKE7K3YA6IP","created_at":"2026-07-05T09:46:42.695738+00:00"},{"alias_kind":"pith_short_8","alias_value":"BE7ARHKE","created_at":"2026-07-05T09:46:42.695738+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/BE7ARHKE7K3YA6IPC2YNRFVSTF","json":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF.json","graph_json":"https://pith.science/api/pith-number/BE7ARHKE7K3YA6IPC2YNRFVSTF/graph.json","events_json":"https://pith.science/api/pith-number/BE7ARHKE7K3YA6IPC2YNRFVSTF/events.json","paper":"https://pith.science/paper/BE7ARHKE"},"agent_actions":{"view_html":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF","download_json":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF.json","view_paper":"https://pith.science/paper/BE7ARHKE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2308.14028&json=true","fetch_graph":"https://pith.science/api/pith-number/BE7ARHKE7K3YA6IPC2YNRFVSTF/graph.json","fetch_events":"https://pith.science/api/pith-number/BE7ARHKE7K3YA6IPC2YNRFVSTF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF/action/storage_attestation","attest_author":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF/action/author_attestation","sign_citation":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF/action/citation_signature","submit_replication":"https://pith.science/pith/BE7ARHKE7K3YA6IPC2YNRFVSTF/action/replication_record"}},"created_at":"2026-07-05T09:46:42.695738+00:00","updated_at":"2026-07-05T09:46:42.695738+00:00"}