from the construction of $M_n$ as a Hrushovski limit. Since we have just seen that for any type $\operatorname{tp}(a/A)$ there is a finite set $A_0$ such that ${
 a\mathop{\mathpalette{
 \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
 \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
}{}}_{A_0}A
}$ this shows directly (without counting types) that $T_n$ is superstable.

Using this description of forking it is easy to give a list of regular types such that any nonalgebraic type is non-orthogonal to one of these. This is entirely similar to the list given in [@BP] and we omit the details but will return to this point in Section \[sec:types\]. It is also clear from this description of forking that the geometry on these types is trivial.

Ampleness
=========

We now recall the definition of a theory being $n$-ample:

A theory $T$ eliminating imaginaries is called $n$-ample if possibly after naming parameters there are tuples $a_0,\ldots a_n$ in $M$ such that the following holds:

1.  for $i=0,\ldots n-1$ we have $$\operatorname{acl}(a_0,.\ldots a_{i-1},a_i)\cap \operatorname{acl}(a_0,.\ldots a_{i-1},a_{i+1})=\operatorname{acl}(a_0,.\ldots a_{i-1});$$

2.  ${
     a_n\mathop{\mathpalette{
     \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\mathchardef
     \nn=12854\hss$\nn$\kern1.4\wd0\hss}\hbox to 0pt{\hss$\mid$\hss}\lower.9\ht0
     \hbox to 0pt{\hss$\smile$\hss}\kern\wd0
    }{}}_{}a_0
    },$ and

3.  ${
     a_n\mathop{\mathpalette{
     \setbox0=\hbox{$x$}\kern\wd0\hbox to 0pt{\hss$
     \mid$\hss}\lower.9\ht0\hbox to 0pt{\hss$\smile$\hss}\kern\wd0
    }{}}_{a_i}a_0\ldots a_i
    }$  for $i=0,\ldots n-1$.

\[T:n-ample\] The theory $T_n$ is $n$-ample and any 