{"paper":{"title":"Uniform intersecting families with large covering number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2021-06-09T19:21:00Z","abstract_excerpt":"A family $\\mathcal F$ has covering number $\\tau$ if the size of the smallest set intersecting all sets from $\\mathcal F$ is equal to $\\tau$. Let $M(n,k,\\tau)$ stand for the size of the largest intersecting family $\\mathcal F$ of $k$-element subsets of $\\{1,\\ldots,n\\}$ with covering number $\\tau$. It is a classical result of Erd\\H os and Lov\\'asz that $M(n,k,k)\\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\\tau)$ for $n<k^2$ and large $\\tau$. The results are quite surprising: For example, we show that\n  $M(n,k,\\tau) =(1-o(1)){n-1\\choose k-1}$, if $n = \\lfloor k^{3/"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.05344","kind":"arxiv","version":3},"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/2106.05344/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"}