${\text{VC}}_{\ell }$-dimension and the jump to the fastest speed of a hereditary $\mathcal{L}$-property

Abstract

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 $\{1,\dots ,n\}$. It was previously known there exists a gap between the fastest possible speed of a hereditary $\mathcal{L}$-property and all lower speeds, namely between the speeds $2^{\Theta (n^r)}$ and $2^{o(n^r)}$. We strengthen this gap by showing that for any hereditary $\mathcal{L}$-property $\mathcal{H}$, either $|{\mathcal{H}}_n|=2^{\Theta (n^r)}$ or there is $\epsilon >0$ such that for all large enough $n$, $|{\mathcal{H}}_n|\le 2^{n^{r-\epsilon }}$. This improves what was previously known about this gap when $r\ge 3$. Further, we show this gap can be characterized in terms of ${\text{VC}}_{\ell }$-dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as $\ell$-dependence.

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