Non-trivial 3-wise intersecting uniform families

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 $\frac{2}{5}+\epsilon <\frac{k}{n}<\frac{1}{2}-\epsilon$ then the maximum size is $4\left(\genfrac{}{}{0px}{}{n-4}{k-3}\right)+\left(\genfrac{}{}{0px}{}{n-4}{k-4}\right)$.