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Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with$q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \inE\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We aregoing to prove that if $|E| \ge Cq^{\frac{11}{4}}$, with a sufficiently largeconstant $C>0$, then the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$is equal to $3$. In particular, this means that for sufficiently large subsetsof ${\Bbb F}_q^3$, the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$is the same as the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t({\BbbF}_q^3)$. In some sense the proof leads us to consider the most complicatedpossible configuration that can always be embedded in subsets of ${\Bbb F}_q^3$of size $\ge Cq^{\frac{11}{4}}$.

Dot products in ${\Bbb F}_q^3$ and the Vapnik-Chervonenkis dimension