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](https://mathoverflow.net/questions/78595/a-curious-generalization-of-hellys-theorem).



Two-point Helly
This is true for special families of convex sets, for example axis parallel rectangles, but it is false for general convex sets, even if $6$ is replaced by any other finite number. This was shown by M. Katchalski and D. Nashtir in the paper *On a conjecture of Danzer and Grünbaum* (Proc. Amer. Math. Soc. 124 (1996), 3213-3218, doi: [10.1090/S0002-9939-96-03806-3](https://doi.org/10.1090/S0002-9939-96-03806-3)).


