{"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 $s=2$. Our result is proved in the more general setting of linear hyperg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14634","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/2411.14634/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"}