$M_n$ choose an $\omega$-saturated $M'\equiv M$. By Lemma \[l:K-sat\], $M'$ is $\K_n$-saturated. Now $M'$ and $M_n$ are partially isomorphic and therefore elementarily equivalent.

We will see in Section \[sec:building\] that $T_n$ is the theory of the building of type $A_{\infty, n+1}$ with infinite valencies.

Following *[@BP]* we call a subset $A$ of a model $M$ of $T_n$ *nice* if

1.  any $E_i$-path between elements of $A$ lies entirely in $A$ and

2.  if $a,b\in A$ are connected by a path in $M$ there is a path from $a$ to $b$ inside $A$.

\[l:strong=nice\]\[r:qftp\] Note that a subset $A$ of $M_n$ is strong in $M_n$ if and only if it is nice. (This follows immediately from the definition of strong extension.)

Since $M_n$ is homogeneous for strong subsets, for any nice subset of $M_n$ the quantifier-free type determines the type, i.e. if $\bar a,\bar b$ are nice in $M_n$ and such that $\operatorname{qftp}(\bar a)=\operatorname{qftp}(\bar b)$, then $\operatorname{tp}(\bar a)=\operatorname{tp}(\bar b)$.

We now work in a very saturated model $\overline{M}$ of $T_n$.

\[l:strongext\] If $A$ is a finite set, there is a nice finite set $B$ containing $A$.

Since single vertices are nice it suffices to prove the following

[**Claim:**]{} *If $A$ is 