All Questions

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...

Is there a "right" notion of a projective plane over a general (unital, non-division) ring? Let me explain what type of object I am looking for. Let $R$ be an arbitrary (not necessarily ...

An incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition ...

Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L(...

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...