dimension $1$ is a free pseudoplane, i.e. an $L_1$-graph which does not contain any cycles and such that any vertex has infinitely many neighbours.

Assume that a free pseudospace of dimension $n-1$ has been defined. Then a of dimension $n$ is an $L_n$-graph such that the following holds:

1.  1.  The set of vertices of type $V_0\cup \ldots \cup V_{n-1}$ is a .

    2.  The set of vertices of type $V_1\cup \ldots \cup V_n$ is a free pseudospace of dimension $(n-1)$.

2.  1.  For any vertex $x$ of type $V_0$, $R(x)$ is a free pseudospace of dimension $(n-1)$.

    2.  For any vertex $x$ of type $V_n$, $R(x)$ is a free pseudospace of dimension $(n-1)$.

3.  1.  Any two vertices $x$ and $y$ intersect in some vertex $z$ or the emptyset.

    2.  Any two vertices $x$ and $y$ generate some vertex $z$ or the emptyset.

4.  1.  If $a$ is a vertex of type $V_n$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$, then there is an $E_1\cup\ldots \cup E_{n-1}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_0$-vertices appear in $\gamma$.

    2.  If $a$ is a vertex of type $V_0$ and 