{"paper":{"title":"A Variant of the VC-dimension with Applications to Depth-3 Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Navid Talebanfard, Peter Frankl, Svyatoslav Gryaznov","submitted_at":"2021-11-18T13:04:55Z","abstract_excerpt":"We introduce the following variant of the VC-dimension. Given $S \\subseteq \\{0, 1\\}^n$ and a positive integer $d$, we define $\\mathbb{U}_d(S)$ to be the size of the largest subset $I \\subseteq [n]$ such that the projection of $S$ on every subset of $I$ of size $d$ is the $d$-dimensional cube. We show that determining the largest cardinality of a set with a given $\\mathbb{U}_d$ dimension is equivalent to a Tur\\'an-type problem related to the total number of cliques in a $d$-uniform hypergraph. This allows us to beat the Sauer--Shelah lemma for this notion of dimension. We use this to obtain sev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.09671","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/2111.09671/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"}