a dense flag in $A$, add a vertex of the same type as $y$ to $A$ which is connected to both $x$ and $z$.

We write $A\leq B$ if $B$ arises from $A$ by finitely many elementary strong extensions.

Let $\K_n$ be the class of finite $L_n$-graphs $A$ such that the following holds

1.  $A$ does not contain any $E_i$-cycles for $i=1,\ldots n$.

2.  If $a\neq a'$ are in $A$, they intersect in a vertex of $A$ or the emptyset.

3.  If $a\neq a'$ are in $A$, they generate a vertex of $A$ or the emptyset.

4.  If $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i-1}$, and $\gamma=(a,b,\ldots , b',a)$ is an $E_i\cup E_{i-j}$ path of length $k$, then there is some $E_{i-1}\cup \ldots E_{i-j}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ with all $V_{i-j}$-vertices occurring in $\gamma$.

5.  If $(b,a,b')$ is a path with $a\in V_i,b,b'\in V_{i+1}$, and $\gamma=(a,b,\ldots , b',a)$ is an $E_i\cup\ldots\cup E_{i+j}$ path of length $k$, then there is some $E_{i+1}\cup\ldots \cup E_{i+j}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ with all $V_{i+j}$-vertices occurring in $\gamma$.

Note that $\K_n$ is closed under finite substructures. We next show that 