Sauer's lemma, a well-known result in computational complexity theory, learning theory, and combinatorics, states the following:

Let $\Phi$ be a collection of subsets of a set $U$, and assume that for all $d$-element subsets $x_1, \dots, x_d$ of $U$ we have $$ \Big| S \cap \{x_1, \dots, x_d\}:~~ S \in \Phi\} \Big| < 2^d. $$ (That means, $\Phi$ is of Vapnik-Chervonenkis dimension at most $d$.)

Then for all $x_1, \dots, x_n \in U$ we have $$ \Big| S \cap \{x_1, \dots, x_n\}:~~S \in \Phi \Big| \leq \sum_{i=0}^d \binom{n}{i} \leq \left( \frac{en}{d} \right)^d. $$

Question: Let $m$ be a fixed positive integer. Assume that $\Phi$ is of VC dimension at most $d$. Let $x_1, \dots, x_n \in U$. Furthermore, let $\mathcal{A}$ denote a family of sets which has the property that

1. Every set $A \in\mathcal{A}$ is of the form $S \cap \{x_1, \dots, x_d\}$ for some $S \in \Phi$.
2. For two sets $A_1,A_2$ we have $\big| A_1 \triangle A_2 \big| \geq m.$

What is a good upper bound for the maximal cardinality of $\mathcal{A}$?

In words, what I am looking for is a version of Sauer's lemma under the additional assumption that not all possible sets obtained by intersection are counted, but only those which are "separated" in the sense of the symmetric difference being not too small. (Sauer's lemma is the special case $m=1$.) The application which I have in mind requires $m \approx \varepsilon n$ for some small $\varepsilon$, but I think it is in general an interesting question.