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

Abstract

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

14 pages. Minor updates