{"paper":{"title":"Improved bounds on the maximum diversity of intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Peter Frankl","submitted_at":"2023-04-20T15:07:37Z","abstract_excerpt":"A family $\\mathcal{F}\\subset \\binom{[n]}{k}$ is called an intersecting family if $F\\cap F'\\neq \\emptyset$ for all $F,F'\\in \\mathcal{F}$. If $\\cap \\mathcal{F}\\neq \\emptyset$ then $\\mathcal{F}$ is called a star. The diversity of an intersecting family $\\mathcal{F}$ is defined as the minimum number of $k$-sets in $\\mathcal{F}$, whose deletion results in a star. In the present paper, we prove that for $n>36k$ any intersecting family $\\mathcal{F}\\subset \\binom{[n]}{k}$ has diversity at most $\\binom{n-3}{k-2}$, which improves the previous best bound $n>72k$ due to the first author. This result is de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.11089","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/2304.11089/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"}