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

TL;DR

This paper explores the relationship between the growth rates of hereditary classes of finite structures and a generalized VC-dimension, revealing a sharp dichotomy in their possible speeds based on model-theoretic properties.

Contribution

It establishes a stronger gap in the growth rates of hereditary properties and characterizes this gap using the concept of VC_{ell}-dimension, linking combinatorial and model-theoretic perspectives.

Findings

Identifies a dichotomy in the speed of hereditary properties: either exponential in n^r or significantly slower.

Strengthens previous results on the gap for properties with arity r ≥ 3.

Connects the growth rate gap to VC_{ell}-dimension and model-theoretic dividing lines.

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.