$\text{VC}_{\ell}$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 $\{1,\ldots, n\}$. 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 $n$, $\mathcal{H}_n\leq 2^{n^{r\epsilon}}$. This improves what was previously known about this gap when $r\geq 3$. 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.00470
 arXiv:
 arXiv:1701.00470
 Bibcode:
 2017arXiv170100470T
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics