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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.

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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).

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