 Solution

For each $A\in \mathcal{A}$, define over ${\mathbb{F}}_p$Replace every power $x_i^r$ with $r\ge 1$ 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 $B\in \mathcal{A}$,This is zero when $A\ne B$, whereasThe evaluation matrix is diagonal with nonzero diagonal, so the polynomials $P_A$ are linearly independent. The space of multilinear polynomials of degree at most $m$ has the monomial basis ${\prod }_{i\in S}{x}_i$ for $|S|\le m$ and dimension ${\sum }_{i=0}^m\left(\genfrac{}{}{0ex}{}{n}{i}\right)$. Hence the modular intersection bound for a set family gives