Multidimensional exact classes, smooth approximation and bounded 4-types

TL;DR

This paper introduces and studies multidimensional exact classes, showing that classes of finite structures with bounded 4-types are polynomial exact classes, thus providing precise size measurements of definable sets.

Contribution

It proves that classes of finite structures with at most d 4-types form polynomial exact classes, confirming a conjecture and extending the theory of exact classes.

Findings

Classes with at most d 4-types are polynomial exact classes.

The result confirms a conjecture by Macpherson.

Uses smooth approximation and Lie coordinatisation techniques.

Abstract

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ($R$-mec), a special kind of multidimensional asymptotic class ($R$-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatisation [14] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language $\mathcal{L}$ and any positive integer $d$ the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types is a polynomial exact class in $\mathcal{L}$, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.