{"paper":{"title":"Maximal $3$-wise intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund, Casey Tompkins, Ce Chen, Haoran Luo, J\\'ozsef Balogh, Kevin Hendrey, Tuan Tran","submitted_at":"2021-10-25T07:40:16Z","abstract_excerpt":"A family $\\mathcal{F}$ on ground set $[n]:=\\{1,2,\\ldots, n\\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\\mathcal{F}$ has non-empty intersection, and no other set can be added to $\\mathcal{F}$ while maintaining this property. In 1974, Erd\\H{o}s and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. We answer their question for $k=3$ and sufficiently large $n$. We show that the unique minimum family is obtained by partitioning the ground set $[n]$ into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.12708","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/2110.12708/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"}