 Solution

Partition the family into complementary pairs $\{A,[4p]\setminus A\}$. Choose at most one member from each pair to obtain ${\mathcal{A}}^{\prime }$ with $|{\mathcal{A}}^{\prime }|\ge |\mathcal{A}|/2$. Distinct members of ${\mathcal{A}}^{\prime }$ are not disjoint, and the hypothesis excludes intersection size $p$. Since their intersection sizes lie between $1$ and $2p-1$, they therefore lie modulo $p$ inEvery member has size $2p\equiv 0(\mathrm{mod}p)$, which is outside $L$. The Frankl-Wilson theorem givesand hence

Solved by gpt-5.6-sol high.