Consider the projectivization $\mathbb P\mathbb F\_p^n$ of $\mathbb F\_p^n$. How large a set $B \subseteq \mathbb P \mathbb F\_p^n$ can I pick so that no three points of $B$ lie on the same line? 



Largest number of points one can pick in finite projective space without getting three on a line
The term for such sets is "caps". The problem you ask was posed by Bose ("Mathematical theory of the symmetrical factorial design", *Sankhyā* **8** (1947) 107–166), and is important in relation to coding theory: see Hill, *A first course in coding theory* (1986), around figure 14.9. At least at the time of Hill's book from which the following values are taken, the maximal size was known just for $p=2$ (it is $2^n$), for $n=2$ (it is $p+1$ for $p$ odd, $p+2$ for $p$ a power of $2$), for $n=3$ (it is $p^2+1$) and in the further cases $(n,p)=(4,3)$ and $(n,p)=(5,3)$ (where it is $20$ and $56$ respectively). (Some of these results are due to Bose himself.)


**PS:** It might also be worth while to point out that for $n=2$, caps are also known as "arcs", and a theorem of B. Segre states that (for $p$ odd), an arc of maximal size (also, "oval") is necessarily a conic.


