Let $T$ be a first order theory. Consider the following finiteness property for $T$:
If $M$ is a finite model for $T$, then the number of definable subsets of $M^n$ is bouned by a number $s(n)$ which depends on $n$ but not on $M$. Moreover, there are first order formulas $\phi_1,\ldots \phi_{s(n)}$ in the first order language of $T$ such that all the definable subsets of $M^n$ are of the form $\{(m_1,\ldots m_n)| \phi_i(m_1,\ldots,m_n)\}$ for some $i=1,\ldots s(n)$.
An example for such a theory is the theory of vector spaces over a finite field $k$. A slightly more general example is the theory of abelian groups of bounded rank. The theory of abelian groups or of groups will be a counter example, since the number of subsets $\{m\in M| m\text{ has order }n\}$ for different $n\in \mathbb{Z}$ is unbounded (for an arbitrary finite abelian group). Another counterexample is the theory of ordered sets, since the number of subsets $\{m\in M|\exists x_1,\ldots x_n : x_1<x_2<\cdots<x_n<m\}$ is unbounded for an arbitrary ordered set. My question is if this property was studied, and if there are any general things that can be said about such theories, or if they perhaps can be classified.
