= Fixed-core construction avoiding midpoint intersections
{title2=$|\mathcal A|=2\binom{3p-1}{p-1}$}
Fix a $(p+1)$-set $B\subseteq[4p]$. Take every $2p$-set $B\cup X$ with $X$ a $(p-1)$-subset of the remaining $3p-1$ points, together with the complements of all these sets. Within either half, intersections have size at least $p+1$. Across the halves they have size $|X\setminus Y|\leq p-1$. Thus no intersection has size $p$, and the two disjoint halves give the displayed family size.
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