have $x\in f_{i+1}$ or $x$ occurs in $\gamma$ between an element of $f_i$ and an element of $f_{i+1}$. By Corollary \[r:flagcl\] we have $x\in\operatorname{acl}(f_if_{i+1})\cap\operatorname{acl}(f_if_{i+2})$. Then $${
 x\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{f_i}a_0a_1,\ldots a_iA
},$$ so $x\notin\operatorname{acl}(a_0a_1,\ldots a_iA)$, a contradiction.

The proof shows that in fact the following stronger ampleness result holds:

If $a_0,\ldots a_n$ are witnesses for $T_n$ being $n$-ample, then there are vertices $b_i\in\operatorname{acl}(a_i)$ such that $(b_0,\ldots b_n)$ is a flag.

Buildings and the prime model of $T_n$ {#sec:building}
======================================

We now turn towards constructing the prime model $M_n^0$ of $T_n$ as a Hrushovski-limit. We will show that $M_n^0$ is the building associated to a right-angled Coxeter group.

For this purpose we now consider an expansion $L_n'$ of the language $L_n$ by binary function symbols $f^i_k$. For an $L_n$-graph $A$ we put $f^i_k(x,y)=z$ if $z$ is the $k^{th}$ element on a unique shortest $E_i$-path of length at least $k$ from $x$ to $y$ and $z=x$ otherwise.

We say that an $L_n$-graph $A$ is $E_i$-connected if the set $V_{i-1}(A)\cup V_i(A)$ is connected.

Let $\K'$ be the class of finite $L_n'$-graphs $A\in\K$ which are $E_i$-connected for $i=1,\ldots n$ and additionally satisfy the following condition:

-   If $a\in A$ is 