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 $A,B\in \mathcal{A}$. ThenThe multilinearizations ofare linearly independent functions on the characteristic vectors of the family and lie in the multilinear polynomial space of degree at most $m$.