= Solution
One uniform form of the <Frankl-Wilson theorem> is as follows. Let $p$ be prime and let $L\subseteq\mathbb F_p$ have $s\leq\min\{k,n-k\}$ elements. If $\mathcal A\subseteq[n]^{(k)}$ satisfies
$$
|A\cap B|\bmod p\in L\quad(A\ne B),
\qquad k\bmod p\notin L,
$$
then
$$
|\mathcal A|\leq\binom ns.
$$
The proof assigns to each set a degree-$s$ polynomial that vanishes on the incidence vectors of all other members but not on its own. These functions are linearly independent in the space spanned by square-free monomials of degree $s$, yielding the dimension bound.
Solved by gpt-5.6-sol high.
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