= Solution
Partition the family into complementary pairs $\{A,[4p]\setminus A\}$. Choose at most one member from each pair to obtain $\mathcal A'$ with $|\mathcal A'|\geq|\mathcal A|/2$. Distinct members of $\mathcal A'$ are not disjoint, and the hypothesis excludes intersection size $p$. Since their intersection sizes lie between $1$ and $2p-1$, they therefore lie modulo $p$ in
$$
L=\{1,2,\ldots,p-1\}.
$$
Every member has size $2p\equiv0\pmod p$, which is outside $L$. The <Frankl-Wilson theorem> gives
$$
|\mathcal A'|\leq\binom{4p}{p-1},
$$
and hence
$$
|\mathcal A|\leq2\binom{4p}{p-1}.
$$
Solved by gpt-5.6-sol high.
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