Happy Ending problem

Happy Ending problem
David Eppstein at English Wikipedia · Public domain · source
Name	Happy Ending problem
Field	Combinatorics, Discrete geometry
Introduced	1933
Contributors	Paul Erdős, George Szekeres, Esther Klein
Notable results	Erdős–Szekeres theorem, Ramsey theory, Erdős–Moser problem

Happy Ending problem

The Happy Ending problem is a classical question in Combinatorics and Discrete geometry about finding convex polygons among points in the plane; it was posed in 1933 and inspired a body of work linking Paul Erdős, George Szekeres, and Esther Klein to fundamental developments in Ramsey theory and extremal combinatorics. The problem launched studies that connected to results by Erdős–Szekeres, influenced research by mathematicians associated with Bell Labs, Harvard University, and Cambridge University, and motivated later investigations by scholars at institutions such as Princeton University, University of Cambridge, and Stanford University.

History and origin

The origin traces to a 1933 conversation among George Szekeres, Esther Klein, and Paul Erdős at a gathering in Cambridge where Klein drew attention to a puzzle about points in general position: her anecdote led Szekeres to formulate a theorem and publish a short proof. Szekeres and Klein later corresponded with Erdős, whose prolific collaboration network across Jerusalem, Budapest, and Princeton amplified visibility of the question; the name "Happy Ending" derives from the marriage of Szekeres and Klein, echoing titles like Love in the Time of Cholera in popular etymology. Early dissemination through seminars at Trinity College, Cambridge and lectures at University of Melbourne spread interest among combinatorialists such as those in the circles of R. L. Graham and Ronald Graham’s collaborators.

Statement and examples

The classical statement asks: given n points in the plane in general position (no three collinear), must one always find a subset of k points that are the vertices of a convex k-gon? Szekeres and colleagues proved affirmative existence for k=4 and k=5, formalized in the Erdős–Szekeres theorem which guarantees a convex k-gon within N(k) points. Examples illustrating small cases include placing points on a convex curve like the boundary of a convex polygon studied by Paul Erdős and arranging points in grid-like patterns examined by researchers at University of Illinois Urbana–Champaign and ETH Zurich. Counterexamples to stronger naive statements often reference constructions developed by mathematicians such as Horton, whose constructions appeared in discussions at American Mathematical Society meetings.

Generalizations and variants

Numerous generalizations consider colored points, higher dimensions, and relaxed position assumptions; these link to work in Ramsey theory and problems tackled by researchers at Massachusetts Institute of Technology and California Institute of Technology. Colored variants ask for monochromatic convex k-gons and relate to the Gallai–Hasse–Roy–Vitaver theorem in spirit, while higher-dimensional analogues investigate convex polytopes studied by scholars at École Polytechnique and Max Planck Institute for Mathematics. Algorithmic and computational variants connect to complexity questions investigated at Carnegie Mellon University and University of Toronto, and probabilistic generalizations have been addressed by probabilists at Kolmogorov Institute-affiliated groups and by combinatorialists collaborating with Microsoft Research.

Proofs and key results

Szekeres and Erdős established bounds for N(k) by combinatorial and geometric arguments; the original proof used the pigeonhole principle in the style familiar from Paul Erdős’s methods and was refined by later authors including R. L. Graham and Ronald Graham collaborators who applied enumerative techniques. Key results include exact values of N(k) for small k, asymptotic bounds showing N(k) is at most exponential in k, and lower bounds constructed via extremal configurations inspired by Horton sets and variations developed in seminars at Institute for Advanced Study. Significant progress involved techniques from Dilworth's theorem-style partial order theory, applications of the Erdős–Szekeres monotone subsequence theorem, and combinatorial geometry methods employed by researchers at Princeton University and University of Oxford.

Applications and related problems

The problem has direct implications for computational geometry tasks pursued at Google Research and IBM Research, such as convex hull algorithms and pattern recognition used in computer vision groups at Carnegie Mellon University. Related problems include the study of empty convex polygons, addressed by teams at National University of Singapore and University of Waterloo, and connections to Ramsey theory problems explored by researchers affiliated with University of Chicago and Rutgers University. Further ties reach into tiling problems and discrete optimization studied at INRIA and Tokyo Institute of Technology, while pedagogical expositions appear in collections from American Mathematical Society and texts used in courses at University of California, Berkeley.

Category:Combinatorics