= Solution
For each $A\in\mathcal A$, define over $\mathbb F_p$
$$
P_A(x_1,\ldots,x_n)
=\prod_{e\in E}\left(\sum_{i\in A}x_i-e\right).
$$
Replace every power $x_i^r$ with $r\geq1$ by $x_i$; this does not change the values on <characteristic vectors of sets> and produces a multilinear polynomial of degree at most $m$.
At the characteristic vector of $B\in\mathcal A$,
$$
P_A(\mathbf1_B)=\prod_{e\in E}(|A\cap B|-e).
$$
This is zero when $A\ne B$, whereas
$$
P_A(\mathbf1_A)=\prod_{e\in E}(|A|-e)\ne0.
$$
The evaluation matrix is diagonal with nonzero diagonal, so the polynomials $P_A$ are <linearly independent vectors>[linearly independent]. The space of multilinear polynomials of degree at most $m$ has the monomial basis $\prod_{i\in S}x_i$ for $|S|\leq m$ and dimension $\sum_{i=0}^m\binom ni$. Hence the <modular intersection bound for a set family> gives
$$
\boxed{|\mathcal A|\leq\binom n0+\binom n1+\cdots+\binom nm}.
$$
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