 Solution

The required prime-power form of the Frankl-Wilson theorem is: if $A_1,\dots ,A_m\subseteq [n]$, no $|A_i|$ is divisible by $p^r$, and every intersection of $k$ distinct members has size divisible by $p^r$, thenIt 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 boundTherefore a family of size $(k-1)(n+1)$ does not exist.