{"paper":{"title":"Recursive lower bounds for uniform set systems of bounded VC-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.CO","authors_text":"Gennian Ge, Xiaochen Zhao","submitted_at":"2026-06-20T14:26:23Z","abstract_excerpt":"For integers $n\\ge d+1$, let $\\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \\[\n  \\mathsf{M}_d(n)\\ge \\binom{n-1}{d}+\\binom{n-4}{d-2}. \\] We introduce a two-cover lifting construction and prove the recursive lower bound \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4}{d-2}+\\mathsf{M}_{d-3}(n-5) \\] for every $d\\ge 3$ and $n\\ge d+3$. Consequently, \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22064","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/2606.22064/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"}