On the maximum relational complexity of affine groups

Let $G$ be a group of permutations of a set $\mathrm{\Omega }$. The relational complexity $\mathrm{R}\mathrm{C}(G)$ is the least integer $r\ge 2$ such that the $G$-orbit of every finite tuple is determined by the $G$-orbits of its $r$-subtuples; there is currently only a small collection of groups $G$ for which $\mathrm{R}\mathrm{C}(G)$ is known precisely. In this paper we study the relational complexity of finite affine groups acting naturally on a $d$-dimensional vector space over ${\mathbb{F}}_q$. We prove that every such group has relational complexity at most $d+2$ when $q>2$ and at most $d+1$ when $q=2$; we produce a complete classification of the primitive such groups for which these bounds are attained.