One-dimensional asymptotic classes of finite structures

A collection ${\mathcal C}$ of finite $\mathcal{L}$-structures is a 1-dimensional asymptotic class if for every $m \in {\mathbb{N}}$ and every formula $\varphi(x,\bar{y})$, where $\bar{y}=(y_1,\ldots,y_m)$:

(i)

There is a positive constant $C$ and a finite set $E\subset {\mathbb{R}}^{>0}$ such that for every $M\in {\mathcal C}$ and $\bar{a}\in M^m$, either $\vert\varphi(M,\bar{a})\vert\leq C$, or for some $\mu\in E$,

$\big\vert\vert\varphi(M,\bar{a})\vert-\mu \vert M\vert\big\vert \leq C\vert M\vert^{\frac{1}{2}}.$

(ii)

For every $\mu\in E$, there is an $\mathcal{L}$-formula $\varphi_{\mu}(\bar{y})$, such that $\varphi_{\mu}(M^m)$ is precisely the set of $\bar{a}\in M^m$ with

$\big\vert\vert\varphi(M,\bar{a})\vert-\mu \vert M\vert\big\vert \leq C\vert M\vert^{\frac{1}{2}}.$

One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.