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Suppose that for every $s$-set $S$ in $\\mathcal{P}$, there is a pair of points in $S$ that lies in a line from $\\mathcal{L}$. We prove that $|\\mathcal{L}| \\ge (n-1)/(s-1)+s-1$ for $n$ large, and this is sharp when $n-1$ is a multiple of $s-1$. This generalizes the de Bruijn-Erd\\H os theorem which is the case $s=2$. Our result is proved in the more general setting of linear hyperg"},"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":"2411.14634","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-21T23:55:35Z","cross_cats_sorted":[],"title_canon_sha256":"ee88d786cf0f40b3c0c0b6117f745aff2931e8edf0e7a08dfe52fc422b1c0e6e","abstract_canon_sha256":"586f09b2766573809c0ff795a33164a9e09e7b952abd4cd2e945eb372936ea59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:38:54.438243Z","signature_b64":"NZXJIl7cB2+erK7YU+ACnilodLfS6LQD6n0NApKg+XKANFz9Q9lRdtzEsuRh1xGFynrBP1xDf8rebsgwNuGGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20610bd072738b4ba1860361a9ce4c2122da62b3042692920b0dd50cf0f972cc","last_reissued_at":"2026-07-05T09:38:54.437425Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:38:54.437425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combining the theorems of Tur\\'an and de Bruijn-Erd\\H os","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Sayok Chakravarty","submitted_at":"2024-11-21T23:55:35Z","abstract_excerpt":"Fix an integer $s \\ge 2$. Let $\\mathcal{P}$ be a set of $n$ points and let $\\mathcal{L}$ be a set of lines in a linear space such that no line in $\\mathcal{L}$ contains more than $(n-1)/(s-1)$ points of $\\mathcal{P}$. Suppose that for every $s$-set $S$ in $\\mathcal{P}$, there is a pair of points in $S$ that lies in a line from $\\mathcal{L}$. We prove that $|\\mathcal{L}| \\ge (n-1)/(s-1)+s-1$ for $n$ large, and this is sharp when $n-1$ is a multiple of $s-1$. This generalizes the de Bruijn-Erd\\H os theorem which is the case $s=2$. 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