if $B\in \K$ and $B'$ is an elementary strong extension of $B$, then also $B'\in\K$.

This shows that the class $(\K,\leq)$ has a Hrushovski limit $M^0_n$. Clearly, $M^0_n$ is $E_i$-connected for $i=1,\ldots n$ and since any two vertices of $M_n^0$ lie in a maximal flag, it follows that $M_n^0$ is in fact $n$-connected. Note that an $L_n'$-substructure of $M_n^0$ is automatically nice, see Remark \[r:qftp\].

The same proof as in the case of $M_n$ shows the first part of the following proposition:

\[p:primmodell\] The Hrushovski limit $M^0_n$ is a model of $T_n$ and $M^0_n$ is the unique countable model of $T_n$ which is $E_i$-connected for $i=1,\ldots n$ and such that every vertex is contained in a maximal flag.

(Note that in [@BP] the corresponding Remark 3.6 of uses Lemma 3.2, which is not correct as phrased there: $M_n^0$ and $M_n^0\cup\{a\}$ with $a$ an isolated point are not isomorphic, but satisfy the assumptions of Remark 3.6.)

The uniqueness part of Proposition \[p:primmodell\] follows directly from the following theorem and Proposition 5.1 of [@HP] which states that this type of building is uniquely determined by its associated Coxeter group and the cardinality of the residues.

\[t:building\] $M^0_n$ is a building of type 