  A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal
$k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$
has non-empty intersection, and no other set can be added to $\mathcal{F}$
while maintaining this property. In 1974, Erd\H{o}s and Kleitman asked for the
minimum size of a maximal $k$-wise intersecting family. We answer their
question for $k=3$ and sufficiently large $n$. We show that the unique minimum
family is obtained by partitioning the ground set $[n]$ into two sets $A$ and
$B$ with almost equal sizes and taking the family consisting of all the proper
supersets of $A$ and of $B$.
