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It is a classical result of Erd\\H os and Lov\\'asz that $M(n,k,k)\\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\\tau)$ for $n<k^2$ and large $\\tau$. The results are quite surprising: For example, we show that\n  $M(n,k,\\tau) =(1-o(1)){n-1\\choose k-1}$, if $n = \\lfloor k^{3/"},"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":"2106.05344","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-06-09T19:21:00Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"8abc9788ec48c5df7371954659888bf9b32d9a6d390dcf733a784bb9fa52992b","abstract_canon_sha256":"ae0bf0e8326d104861d077b592131489949d1bca6460db7c636e1e8ac227fdd6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:07:28.456605Z","signature_b64":"seCKvQcFoh5ME1q503mfEJKmjWk2I77DpRAv4/CFITnpQDEEeFTw7qJy+3qkx5UXgc/LUyXtvdQSuiOdZsY+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1d45a54263b26c0b2bee2659dd737be502d5e57a76718ee0939f22a76cf460e","last_reissued_at":"2026-07-05T06:07:28.456132Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:07:28.456132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform intersecting families with large covering number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2021-06-09T19:21:00Z","abstract_excerpt":"A family $\\mathcal F$ has covering number $\\tau$ if the size of the smallest set intersecting all sets from $\\mathcal F$ is equal to $\\tau$. Let $M(n,k,\\tau)$ stand for the size of the largest intersecting family $\\mathcal F$ of $k$-element subsets of $\\{1,\\ldots,n\\}$ with covering number $\\tau$. It is a classical result of Erd\\H os and Lov\\'asz that $M(n,k,k)\\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\\tau)$ for $n<k^2$ and large $\\tau$. The results are quite surprising: For example, we show that\n  $M(n,k,\\tau) =(1-o(1)){n-1\\choose k-1}$, if $n = \\lfloor k^{3/"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.05344","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.05344/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":"2106.05344","created_at":"2026-07-05T06:07:28.456196+00:00"},{"alias_kind":"arxiv_version","alias_value":"2106.05344v3","created_at":"2026-07-05T06:07:28.456196+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.05344","created_at":"2026-07-05T06:07:28.456196+00:00"},{"alias_kind":"pith_short_12","alias_value":"4HKFUVBGHMTM","created_at":"2026-07-05T06:07:28.456196+00:00"},{"alias_kind":"pith_short_16","alias_value":"4HKFUVBGHMTMBMV6","created_at":"2026-07-05T06:07:28.456196+00:00"},{"alias_kind":"pith_short_8","alias_value":"4HKFUVBG","created_at":"2026-07-05T06:07:28.456196+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/4HKFUVBGHMTMBMV64JSZ3VZXXZ","json":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ.json","graph_json":"https://pith.science/api/pith-number/4HKFUVBGHMTMBMV64JSZ3VZXXZ/graph.json","events_json":"https://pith.science/api/pith-number/4HKFUVBGHMTMBMV64JSZ3VZXXZ/events.json","paper":"https://pith.science/paper/4HKFUVBG"},"agent_actions":{"view_html":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ","download_json":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ.json","view_paper":"https://pith.science/paper/4HKFUVBG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2106.05344&json=true","fetch_graph":"https://pith.science/api/pith-number/4HKFUVBGHMTMBMV64JSZ3VZXXZ/graph.json","fetch_events":"https://pith.science/api/pith-number/4HKFUVBGHMTMBMV64JSZ3VZXXZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ/action/storage_attestation","attest_author":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ/action/author_attestation","sign_citation":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ/action/citation_signature","submit_replication":"https://pith.science/pith/4HKFUVBGHMTMBMV64JSZ3VZXXZ/action/replication_record"}},"created_at":"2026-07-05T06:07:28.456196+00:00","updated_at":"2026-07-05T06:07:28.456196+00:00"}