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Happy Ending Problem

This sketch is inspired by a great story at Pat Bellew's math history blog. The theorem (illustrated here for the smallest case) involves Esther Klein, Paul Erdos and George Szekeres, plus the birth of combinatorial geometry, and the marriage of two mathematicians! This theorem is also covered at Theorem of the Day: The question is: given 5 points in the plane, can you arrange them so any 4 of them form a concave quadrilateral?