## Multidimensional exact classes and Lie coordinatisation

Abstract:

We introduce the notion of a multidimensional exact class, as jointly developed with Anscombe, Macpherson and Steinhorn. We then sketch a proof of the following result: For any countable language $\mathcal{L}$ and for any positive $d\in\mathbb{N}$, the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types forms a multidimensional exact class. The proof makes extensive use of smooth approximation, a notion introduced by Lachlan in the 1980s, and the equivalent notion of Lie coordinatisation, as deeply developed by Cherlin and Hrushovski in their book

We introduce the notion of a multidimensional exact class, as jointly developed with Anscombe, Macpherson and Steinhorn. We then sketch a proof of the following result: For any countable language $\mathcal{L}$ and for any positive $d\in\mathbb{N}$, the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types forms a multidimensional exact class. The proof makes extensive use of smooth approximation, a notion introduced by Lachlan in the 1980s, and the equivalent notion of Lie coordinatisation, as deeply developed by Cherlin and Hrushovski in their book

*Finite Structures with Few Types*(Princeton, 2003).