{"paper":{"title":"Intersecting families with covering number three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Peter Frankl","submitted_at":"2022-07-12T11:55:00Z","abstract_excerpt":"We consider $k$-graphs on $n$ vertices, that is, $\\mathcal{F}\\subset \\binom{[n]}{k}$. A $k$-graph $\\mathcal{F}$ is called intersecting if $F\\cap F'\\neq \\emptyset$ for all $F,F'\\in \\mathcal{F}$. In the present paper we prove that for $k\\geq 7$, $n\\geq 2k$, any intersecting $k$-graph $\\mathcal{F}$ with covering number at least three, satisfies $|\\mathcal{F}|\\leq \\binom{n-1}{k-1}-\\binom{n-k}{k-1}-\\binom{n-k-1}{k-1}+\\binom{n-2k}{k-1}+\\binom{n-k-2}{k-3}+3$, the best possible upper bound which was proved in \\cite{F80} subject to exponential constraints $n>n_0(k)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.05487","kind":"arxiv","version":4},"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/2207.05487/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"}