# Two-point Helly

Suppose that for a finite collection of planar convex sets $$mathcal F$$ the following holds.
For any six members of $$mathcal F$$ there are two points such that every set contains (at least) one of the points.
Does it follow that all members of $$mathcal F$$ can be stabbed by two points?

I am sure that this is known, and probably even the more general problem of determining the optimal value instead of six for $$k$$ stabbing points in $$mathbb R^d$$.
Related question with many links to related problems: A curious generalization of Helly's theorem.