  Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite
field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed
nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where
$h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the
authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq
Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of
$\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary
dimension and improve the exponent in the case $d=3$.

Combinatorics