Maximum number of points in general position in a random subset of finite $3$-dimensional spaces

Abstract: Let $\alpha(\mathbb{F}_q<sup>{d},p)$ be the maximum possible size of a point set in general position in the $p$-random subset of $\mathbb{F}_q<sup>d$. In this note, we determine the order of magnitude of $\alpha(\mathbb{F}_q<sup>{3},p)$ up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of $4$ points in the same plane.