{"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"}