## 311 – Another extra credit exercise

This optional set is due Tuesday, February 24, at the beginning of lecture. Some of these problems are harder than others, you do not need to address all of them. Contact me if you would like additional time to keep working on some of the more difficult ones.

Recall the theories $T_n$ introduced in homework 2:

• For any two distinct points passes a unique line.
• Any line has at least $n$ points.
• There are three non-collinear points.

Recall also the axioms of projective geometry:

1. For any two distinct points passes a unique line.
2. Any two distinct lines meet at a point.
3. There are at least four points, no three of them collinear.

Theorem ($T_{n+1}$). There are at least $n^2+n+1$ points.

Proof. There is a line $\ell$ with at least $n+1$ points $a_1,\dots,a_{n+1}$, and there is at least one point $a$ not in $\ell$. The lines $\overleftrightarrow{aa_1},\dots,\overleftrightarrow{aa_{n+1}}$ are all different, and each of them has at least $n-1$ points other that $a$ and the $a_i$. It is not hard to see that all these points are different, giving us at least: $n+1$ points in $\ell$, $n-1$ points in each of the lines $\overleftrightarrow{aa_i},$ and $a$, that is, $(n+1)+(n-1)(n+1)+1=n^2+n+1$ points. $\Box$

Recall that if $P$ is a finite model of the projective axioms, then there is a natural number $n$ (the order of $P$) such that $P$ has exactly $n^2+n+1$ points and $n^2+n+1$ lines, each point belongs to exactly $n+1$ lines, and each line goes through exactly $n+1$ points. (See here for a reference with proofs of these facts, from Perspectives on projective geometry, by Jürgen Richter-Gebert.)

• Show that the only model of $T_3$ with $7$ points is the Fano plane.
• Show that all finite projective planes of order three are isomorphic (an example is described here).
• Is any model of $T_4$ with $13$ points the finite projective plane of order three?
• Is any model of $T_5$ with $21$ points the finite projective plane of order four? More generally, if $n>1$, is any model of $T_{n+1}$ with $n^2+n+1$ points a finite projective plane of order $n$?