  An $m$-general set in $AG(n,q)$ is a set of points such that any subset of
size $m$ is in general position. A $3$-general set is often called a capset. In
this paper, we study the maximum size of an $m$-general set in $AG(n,q)$,
significantly improving previous results. When $m=4$ and $q=2$ we give a
precise estimate, solving a problem raised by Bennett.
