D}D$, a contradiction.

For any vertex $a$ and set $A$, there is a flag $C\in\operatorname{acl}(A)$ such that for any $b\in\operatorname{acl}(A)$ there is a reduced path from $a$ to $b$ passing through one of the elements of $C$.

The flag $C$ is called the *projection* from $a$ to $A$ and we write $C=\operatorname{proj}(a/A)$. Note that $\operatorname{proj}(a/A)=\emptyset$ if and only if $a$ is not connected to any vertex of $\operatorname{acl}(A)$.

Let $b_1,b_2\in\operatorname{acl}(A)$ and let $\gamma_1=(x_0=a,\ldots ,x_m=b_1)$ and $\gamma_2=(y_0=a,\ldots ,y_l=b_2)$ be reduced paths such that $i,j$ are minimal possible with $x_i,y_j\in\operatorname{acl}(A)$. By composing the initial segments of $\gamma_1$ and $\gamma_2$ and reducing we obtain a reduced path from $x_i$ to $y_j$ intersecting $\operatorname{acl}(A)$ only in $x_i,y_j$ since $x_i, y_j$ were chosen at minimal distance from $a$. By Corollary \[r:flagcl\] $x_i,y_j$ is a flag. Thus the set of such vertices forms a flag $C$.

It is now easy to show the following:

\[T:stable\] The theory $T_n$ is $\omega$-stable.

Let $M$ be a countable model and let $\bar d$ be a tuple from $\overline{M}$. Let $C\in M$ be the finite set of projections from $\bar d$ to $M$. Then the type $\operatorname{tp}(\bar d/M)$ is determined by $\operatorname{tp}(\bar d/C)$. By Lemmas \[l:strong=nice\] and \[l:strongext\], $\bar d\cup 