---
abstract: 'We give a uniform construction of free pseudospaces of dimension $n$ extending work in [@BP]. This yields examples of $\omega$-stable theories which are $n$-ample, but not $n+1$-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.'
author:
- Katrin Tent
title: 'The free pseudospace is $n$-ample, but not $(n+1)$-ample '
---

Introduction
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In the investigation of geometries on strongly minimal sets the notion of ampleness plays an important role. Algebraically closed fields are $n$-ample for all $n$ and it is not known whether there are strongly minimal sets which are $n$-ample for all $n$ and do not interpret an infinite field. Obviously, one way of proving that no infinite field is interpretable in a theory is by showing that the theory is *not* $n$-ample for some $n$.

In [@BP], Baudisch and Pillay constructed a free pseudospace of dimension $2$. Its theory is ${\omega}$-stable (of infinite rank) and $2$-ample. F. Wagner posed the question whether this example was $3$-ample or not.

In Section \[sec:construction\] we give a uniform construction of a free pseudospace of dimension $n$ and show that it is $n$-ample, but not $n+1$-ample. It turns out that the theory of the 