{"paper":{"title":"On set systems without a simplex-cluster and the Junta method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Noam Lifshitz","submitted_at":"2018-04-03T15:10:56Z","abstract_excerpt":"A family $\\{A_{0},\\ldots,A_{d}\\}$ of $k$-element subsets of $[n]=\\{1,2,\\ldots,n\\}$ is called a simplex-cluster if $A_{0}\\cap\\cdots\\cap A_{d}=\\varnothing$, $|A_{0}\\cup\\cdots\\cup A_{d}|\\le2k$, and the intersection of any $d$ of the sets in $\\{A_{0},\\ldots,A_{d}\\}$ is nonempty. In 2006, Keevash and Mubayi conjectured that for any $d+1\\le k\\le\\frac{d}{d+1}n$, the largest family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster is the family of all $k$-subsets that contain a given element. We prove the conjecture for all $k\\ge\\zeta n$ for an arbitrarily small $\\zeta>0$, provid"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01026","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":""},"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"}