= Solution
The required prime-power form of the <Frankl-Wilson theorem> is: if $A_1,\ldots,A_m\subseteq[n]$, no $|A_i|$ is divisible by $p^r$, and every intersection of $k$ distinct members has size divisible by $p^r$, then
$$
m\leq(k-1)(n+1).
$$
It follows by associating to each $A_i$ its incidence vector augmented by a constant coordinate and applying the Frankl-Wilson polynomial independence lemma to the $(k-1)$ layers of multilinear intersection polynomials. The hypotheses make the diagonal evaluations nonzero modulo $p^r$ and every $k$-fold off-diagonal evaluation zero; independence leaves at most $(k-1)(n+1)$ polynomials. Applying the theorem gives the desired bound.
When $r=1$, the argument works directly over $\mathbb F_p$ without the constant-coordinate lift and yields the stronger bound
$$
m\leq(k-1)n.
$$
Therefore a family of size $(k-1)(n+1)$ does not exist.
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