How the ‘happy ending problem’ launched a new branch of math—and a romance
A puzzle about dots on a page led to one of the most profound areas of modern math
In the early 1930s, a group of gifted university students gathered in Budapest to discuss mathematics. Among them was Paul Erdős, who would later become one of the most prolific mathematicians in history, along with George Szekeres and Esther Klein. One day, Klein presented her friends with a seemingly simple puzzle involving dots on a page: can five dots, placed arbitrarily on a page (with no three dots lying on the same straight line), always form the vertices of a convex quadrilateral?
This "happy ending problem" not only launched a new branch of math focused on order within chaos but also sparked a lasting romance between Klein and Szekeres. Klein's elegant proof demonstrated that, no matter how the dots were arranged, four of them would always form a convex quadrilateral. This concept soon piqued Erdős's interest, leading to further research on the problem.
Erdős and Szekeres eventually proved that there exists a finite number of dots required to guarantee the formation of a convex polygon of any desired size, contributing significantly to the field of Ramsey theory.
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