Finite projective planes can be considered as combinatorial ${n^2+n+1}_{n+1}$ configurations. So for example the order 2 projective plane (Fano plane) is a $7_3$ configuration.
It is known that the Fano plane is the only $7_3$ configuration. Likewise, the order 3 projective plane is the unique $13_4$ configuration.
Is it known whether this true for all finite order projective planes?
That is, for each $n$, are all the possible ${n^2+n+1}_{n+1}$ configurations order $n$ projective planes?
Suppose there are two lines that do not intersect. Take the first line that contains $n+1$ points. Each of these points lie on another $n$ distinct lines which cannot be the second non-intersecting line.
Hence the number of lines is at least:
$$(n+1)n+2=n^2+n+2$$
This is a contradiction. So any combinatorial $n^2+n+1_{n+1}$ is a projective plane.