Happy End Problem  
    Esther Klein was a member of the Anonymous group, which met at a bench by the hooded statute of Anonymous in the city park of Budapest, during the Horthy dictatorship in Hungary. Klein once shared with her fellow mathematicians (which included Paul Erdös, György Szekeres, and Endre Makai) a curious problem in plane geometry:
 
    Given any five points on a flat surface, with no three of them in a straight line, prove that four of these points will always form a convex quadrilateral (a four-sided figure with no indentations).
 
    Klein was able to prove the theorem by showing that all conceivable arrangements of five points fell into three general cases, each of which guaranteed a convex quadrilateral.
    The first case is when the five points themselves form a convex pentagon; then any four of the five points form a convex quadrilateral.
    The second case is when the one of the five points lies inside the quadrilateral formed by the other four; then the four exterior points form the convex quadrilateral.
    The remaining case is when two of the five points lie inside the triangle formed by the other three. If a line is drawn through those two points, bifurcating the triangle, two points of the triangle will fall on one side of the line. Those four points form a convex quadrilateral.
   
  Erdös and Szekeres were taken with the elegant proof, so they tried to extend the result to polygons with many more sides. Makai proved that nine points were needed to guarantee a convex pentagon.
    A generalization quickly emerged: Was a convex polygon of n sides guaranteed whenever the points numbered 2^n-2+1? However, they soon had to realize that a simple-minded argument would not do. To Szekeres, the fact that the problem was pointed out by Klein, added a strong incentive; and after a few weeks, he was able to prove the necessity of convex polygons of any given size n, but the proof gave a large value as the number of points required. Nonetheless, Szekeres's accomplishment was impressive enough to win Klein's hand, and the two were married four years later. As a result, Erdös named the problem Happy End Problem. Sixty years later, Esther Klein and György Szekeres are still married.
 
    Erdös soon improved on Szekeres's result, although the gap between the suspected value and Erdös's proven value was still huge: Erdös proved that 71 points were required to guarantee a convex hexagon, although 17 points (17=2⁴+1) were thought to be sufficient.
 
    No progress on the Happy End Problem was made until November 1996. Then, the mathematicians Ronald Graham and Fan Chung (who are married to each other, too), decided to tackle it after having met Szekeres at the Erdös's memorial service. On a flight, they passed the time by working to narrow the gap, and succeeded in lowering the upper bound to 70. As soon as their success became known, Daniel Kleitman succeeded in reducing the guarantee to 65. Then, in the spring of 1997, the guarantee fell to 37. So, progress is still being made, and perhaps another couple is necessary to close the gap to the suspected value of 17.
 
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