In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book *Geometric Stability Theory*.

The facts are stated as follows:

Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally projective, locally finite, infinite homogeneous geometry. Then $(S,\mbox{cl})$ is isomorphic to some affine or projective geometry over a finite field.

and

Fact 2.1.12. If $(S,\mbox{cl})$ is a projective geometry of dimension at least 4, such that all $2$-dimensional closed sets have at least three elements, then $(S,\mbox{cl})$ is a projective geometry over some division ring.

The author states this fact without reference, and I was not able to find one myself. Any directions would be very welcomed.