  A family of $k$-element subsets of an $n$-element set is called 3-wise
intersecting if any three members in the family have non-empty intersection. We
determine the maximum size of such families exactly or asymptotically. One of
our results shows that for every $\epsilon>0$ there exists $n_0$ such that if
$n>n_0$ and $\frac25+\epsilon<\frac kn<\frac 12-\epsilon$ then the maximum size
is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.
