Consider a set $X$ and a collection $\mathcal{H}$ of functions from $X$ to $\{0,1\}$. $\mathcal{H}$ is a finite set $A It says crush \subset X$ . $ to $\{0,1\}$. The Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ is the size of the largest aggregate it crushes. Recent work by Fitzpatrick, Wyman, and his fourth and his seventh authors found a subset of $\F_q^d$, namely the field $\F_q$. For fixed $t \in \F_q^d$, let $h_y(x) = 1$ if $||yx||=t$ and 0 otherwise. = x_1^2 + x_2^2+ \cdots + x_d^2$. For $E\subset \F_q^d$ they have $\Hh_t^d(E) = \{h_y: y \in E\ }$ and asked the magnitude of $E$ to guarantee the next VC dimension. $\Hh_t^d(E)$ is the same as $\Hh_t^d(\F_q^d)$. $|E|\geq Cq^{\frac{15}{8}}$ for $d=2$ for $C$ constant.
In this paper, we examine what happens when we add another parameter to $h_y$ and ask the same question. Specifically, we define $h_{y,z}(x) = 1$ for $y\neq z$ if $||yx|| = ||zx|| $\Hh_t^d Let (E) = \{h_{y,z}: y,z \in E, y\neq z\}$ . Solving this problem in all dimensions, for $d\geq 3$ $|E|\geq Cq^{d-\frac{1}{d-1}}$ ^d(E)$ is $ Same as \Hh_t^d(\F_q^d)$. $d=3$ gives slightly stronger results. This result holds for $|E|\geq Cq^{7/3}$. Moreover, if $d=2$, the result holds when $|E|\geq Cq^{7/4}$. where $C$ is a function of $d$ and is therefore constant with respect to $q$.
