# Looking for reference or proof to some facts stated on Anand Pillay's book

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.

• 2.1.12 is a classical result in projective geometry, any good book should have it. e.g. see math.stackexchange.com/questions/549099/… for a list of books. – Dima Pasechnik May 19 '15 at 11:48
• 2.1.11 is also a result that is very old, at least in geometric terms (I don't know what an "infinite homogeneous" means though). Does it mean that the rank is infinite? – Dima Pasechnik May 19 '15 at 11:54
• These are two separate conditions. It simply means it is infinite as a set, and that for any two points outside a closed set there is a $\mbox{cl}$-automorphism which fixes the set pointwise while mapping one of the points to the other – Shai Deshe May 19 '15 at 11:58
• you need infiniteness to avoid a sporadic example related to the Mathieu group $M_{22}$, I suppose. Well, I don't know how to deal with the case of 2-dimensional closed sets (a.k.a. lines) being of size 2, and locally being an infinite projective plane. – Dima Pasechnik May 19 '15 at 12:10
• if your 2-dimensional closed sets have at least 3 elements then one doesn't need any group actions and infiniteness, it's just pure synthetic geometry to show that you sill get an affine or projective geometry. – Dima Pasechnik May 19 '15 at 13:50