$A_{\infty,n+1}$ all of whose residues have cardinality $\aleph_0$.

Recall the following definitions (see e.g. [@G]).

Let $W$ be the Coxeter group

$$W=\langle t_0,\ldots t_n\colon t_i^2=(t_it_k)^2=1, i,k=0\ldots n, |k-i|\geq 2 \rangle,$$ whose associated diagram we call $A_{\infty,n+1}$.

\[d:building\] A building of type $A_{\infty,n+1}$ is a set $\Delta$ with a *Weyl distance function* $\delta: \Delta^2\to W$ such that the following two axioms hold:

1.  For each $s\in S:=\{t_i,i=0,\ldots n\}$, the relation $x\sim_s y$ defined by $\delta(x,y)\in\{1,s\}$ is an equivalence relation on $\Delta$ and each equivalence class of $\sim_s$ has at least $2$ elements.

2.  Let $w=r_1r_2\ldots r_k$ be a shortest representation of $w\in W$ with $r_i\in S$ and let $x,y\in\Delta$. Then $\delta(x,y)=w$ if and only if there exists a sequence of elements $x,x_0,x_1,\ldots, x_k=y$ in $\Delta$ with $x_{i-1}\neq x_i$ and $\delta(x_{i-1},x_i)=r_i$ for $i=1,\ldots, k$.

A sequence as in 2. is called a *gallery of type* $(r_1,r_2,\ldots, r_k)$. The gallery is called reduced if the word $w=r_1r_2,\ldots, r_k$ is reduced, i.e. a shortest representation of $w$.

We now show how to consider $M_n^0$ as a building of type $A_{\infty,n+1}$.

(of Theorem \[t:building\]) We extend the set of edges of the $n+1$-coloured graph $M_n^0$ by putting edges between any two vertices that are incident 