Finite projective planes How big a set of points in **general position** (i.e., no three collinear) can be found in a finite projective plane of order $n$? I hope the answers won't be too technical, as I know almost nothing about projective planes besides the definition. **Motivation:** I want to use this for an example in graph theory. **My work:** If $S$ is a set of points in general position, then $|S|\le n+2$, since $S$ has at most a point $p$ and another point on each of the $n+1$ lines through $p$. So $n+2$ is an upper bound. For a lower bound, let $S$ be any **maximal** set of points in general position and note that $\binom{|S|-1}2\ge n$, whence $|S|\ge\frac{3+\sqrt{1+8n}}2$.

An _arc_ is a set of points, no three collinear. An arc is _complete_ if it is maximal with respect to set inclusion. So you are asking for the maximum size of a complete arc.

According to this paper, this appears to be a hard problem in general, although if $n$ is even, then your upper bound is attained, and if $n$ is odd, then it can be improved to $n+1$.

I was hoping to track down a proof for the $n$ even case (due to Segre) but it appears to be a bit too obscure. Hopefully you have better luck. Actually it's not clear to me if he "only" proved it for $n=2^k$.