{"paper":{"title":"$\\text{VC}_{\\ell}$-dimension and the jump to the fastest speed of a hereditary $\\mathcal{L}$-property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"C. Terry","submitted_at":"2017-01-02T18:01:06Z","abstract_excerpt":"In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\\text{VC}$-dimension called $\\text{VC}_{\\ell}$-dimension. Let $\\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\\mathcal{L}$-property is a class of finite $\\mathcal{L}$-structures closed under isomorphism and substructures. The \\emph{speed} of a hereditary $\\mathcal{L}$-property $\\mathcal{H}$ is the function which sends $n$ to $|\\mathcal{H}_n|$, where $\\mathcal{H}_n$ is the set of elements of $\\mathcal{H}$ with universe $\\{1,\\ldot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00470","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"}