= Modular intersection bound for a set family
Let $E\subseteq\mathbb F_p$ have size $m$, and let $\mathcal A\subseteq\mathcal P([n])$ satisfy $|A|\notin E$ and $|A\cap B|\in E$ for distinct $A,B\in\mathcal A$. Then
$$
|\mathcal A|\leq\sum_{i=0}^m\binom ni.
$$
The multilinearizations of
$$
P_A(x)=\prod_{e\in E}\left(\sum_{i\in A}x_i-e\right)
$$
are linearly independent functions on the characteristic vectors of the family and lie in the multilinear polynomial space of degree at most $m$.
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