If there are 3 intervals, such that any 2 of them intersect, then all 3 of them intersect.
For any 4 disks, if any 3 of them have a non empty intersection, then all 4 of them have a common intersection.
This seems to generalize to higher dimensions.
What is this property, that for some $n$, knowing all subset of size $n-1$ of $n$ shapes have non-empty intersection implies all the shapes have non-empty intersection?