Let $S$ be a non-empty set. A geometry of type $n$ for $n\geq 1$ on $S$ (consisting of at least $n$ elements) is a set ${\mathfrak P}\subseteq {\mathcal P}(S)$ such that

1. all members of $\mathfrak P$ have at least $n$ elements,

2. any $n$ elements of $S$ are contained in exactly one member of $\mathfrak P$,

3. for $l_1\neq l_2 \in \mathfrak P$ we have $|l_1\cap l_2| = n-1$, and

4. there is $T\subseteq S$ with $|T|=n+1$ and $T\notin \mathfrak P$.

Geometries of type $1$ are "traditional" partitions -- they define an equivalence relation on the set $S$.

A geometry of type $2$ is a projective plane.

Question: Is there for every $n\geq 1$ a geometry $\mathfrak P$ of type $n$ on $\omega$ such that $|\mathfrak P| \geq 2$?