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Let $\\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \\cap \\{1, 2, 3\\}| \\geq 2$. We prove that for $n \\geq 50 k^2$ $|\\mathcal I(\\mathcal F)|$ is maximized by $\\mathcal A$."},"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":"2108.00479","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-08-01T15:41:56Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"0fdb3e63631ffffc017b59c5e476030f8d57b41a2f93666e9da1644aa3595477","abstract_canon_sha256":"faea300b3466a507f57afb1855cfec90c426492c47df31bd16b8a916999206cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:02:23.635303Z","signature_b64":"DFeiULNaYs9bvJBkT1ojdP+usmeYs/+iy5uZo9ir+qcW1oGzrLpgrIl8hcEmiMtHoVkL9ONpufJZjBW2nYFHCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eeb9558a1fe8bad055e59f2015c0051e28ea2365f51a33cc78d764d4602a0359","last_reissued_at":"2026-07-05T03:02:23.634896Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:02:23.634896Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the maximum number of distinct intersections in an intersecting family","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl, Sergei Kiselev","submitted_at":"2021-08-01T15:41:56Z","abstract_excerpt":"For $n > 2k \\geq 4$ we consider intersecting families $\\mathcal F$ consisting of $k$-subsets of $\\{1, 2, \\ldots, n\\}$. Let $\\mathcal I(\\mathcal F)$ denote the family of all distinct intersections $F \\cap F'$, $F \\neq F'$ and $F, F'\\in \\mathcal F$. Let $\\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \\cap \\{1, 2, 3\\}| \\geq 2$. We prove that for $n \\geq 50 k^2$ $|\\mathcal I(\\mathcal F)|$ is maximized by $\\mathcal A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.00479","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/2108.00479/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":"2108.00479","created_at":"2026-07-05T03:02:23.634950+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.00479v1","created_at":"2026-07-05T03:02:23.634950+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.00479","created_at":"2026-07-05T03:02:23.634950+00:00"},{"alias_kind":"pith_short_12","alias_value":"524VLCQ75C5N","created_at":"2026-07-05T03:02:23.634950+00:00"},{"alias_kind":"pith_short_16","alias_value":"524VLCQ75C5NAVPF","created_at":"2026-07-05T03:02:23.634950+00:00"},{"alias_kind":"pith_short_8","alias_value":"524VLCQ7","created_at":"2026-07-05T03:02:23.634950+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/524VLCQ75C5NAVPFT4QBLQAFDY","json":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY.json","graph_json":"https://pith.science/api/pith-number/524VLCQ75C5NAVPFT4QBLQAFDY/graph.json","events_json":"https://pith.science/api/pith-number/524VLCQ75C5NAVPFT4QBLQAFDY/events.json","paper":"https://pith.science/paper/524VLCQ7"},"agent_actions":{"view_html":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY","download_json":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY.json","view_paper":"https://pith.science/paper/524VLCQ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.00479&json=true","fetch_graph":"https://pith.science/api/pith-number/524VLCQ75C5NAVPFT4QBLQAFDY/graph.json","fetch_events":"https://pith.science/api/pith-number/524VLCQ75C5NAVPFT4QBLQAFDY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY/action/storage_attestation","attest_author":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY/action/author_attestation","sign_citation":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY/action/citation_signature","submit_replication":"https://pith.science/pith/524VLCQ75C5NAVPFT4QBLQAFDY/action/replication_record"}},"created_at":"2026-07-05T03:02:23.634950+00:00","updated_at":"2026-07-05T03:02:23.634950+00:00"}