Happy Ending Problems

Theorem. Any set of five points in the plane in general position[1] has a subset of four points that form the vertices of a convex quadrilateral.

The Happy Ending theorem can be proven by a simple case analysis: If four or more points are vertices of the convex hull, any four such points can be chosen. If on the other hand the point set has the form of a triangle with two points inside it, the two inner points and one of the triangle sides can be chosen. See Peterson (2000) for an illustrated explanation of this proof, and Morris & Soltan (2000) for a more detailed survey of the problem than we provide here.

This is still unsolved, maybe mfp can help?