Find $n$ sets with nonempty pairwise intersections, no set a subset of another and $\bigcap_i A_i=\emptyset$ Or, $A_i\cap A_j\ne\emptyset$, $A_i\not\subseteq A_j$ for $i\ne j$, and $\bigcap_i A_i=\emptyset$.
An example for $n=4$:


How about
$$ A_i = \{ \{i,k\} \mid 1\le k\le n \} $$
which works for $n\ge 3$.
$n=2$ is not possible (nonempty pairwise intersections in that case conflicts directly with all of them having empty intersection).
$n=1$ is arguably possible by setting $A_1=\varnothing$, because there are no pairwise intersections to require nonemptiness of.
