Discrete Geometry and The Happy Ending Problem

1. Lindsay Mullen Seminar Presentation #3 November 18, 2013 Discrete Geometry and The Happy Ending Problem • Geometry • Discrete Geometry

2. Discrete Geometry • Closely related to combinatorial geometry • Branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects • Most questions involve: finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth • Focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object

3. Definitions • General Position - no three points are collinear

4. Definitions continued… • Convex Polygon - A polygon is convex if there are no "dents" or indentations in it (no internal angle is greater than 180°)

5. Definitions continued… • Convex hull of X - the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape formed by a rubber band stretched around X

6. The Happy Ending Problem • TheHappy Ending Problemstates: “Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.” • Notation would be f(4)=5 • This was one of the original results that led to the development of Ramsey theory

7. History • 1933– Budapest, Hungary – a group of students known as the Anonymous Society would meet to discuss mathematics • Group consisted of Paul Erdös, Esther Klein, Endre Makai, and György (George) Szekeres • Klein posed the following question after some doodling: “how many points do we need on a plane to guarantee that some four of them form a convex quadrilateral?”

8. History continued… • Erdös, Szekeres, and Klein sought to formalize the result [f(4)=5] and develop a proof for the case with more than 5 points • Klein came up with a proof of different cases for f(4)=5 • Szekeres and Erdösissued a paper, A Combinatorial Problem in Geometry, in 1935 formally proving the result • Specifically generalized a stronger theorem known as the Erdös-Szekeres Conjecture: • “For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon.”

9. History continued… • Erdös dubbed this the Happy Ending Problem because shortly after these results were published,Szekeresand Klein got married • Erdös and Szekeres(1935) claimed EndreMakai proved that f(5)=9, or in words, every set of nine points in the general position contains at least one convex pentagon. • HOWEVER, Makai never published a formal proof so in 1970, J.D. Kalbfleishpublished a proof for this result • 2006 – Szekeres and Lindsay Peters provided a computer proof of f(6)=17, or 17 points in the general position always contains a convex hexagon.

10. Happy Ending Problem continued… • “Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.”

11. Klein’s Proof • Clearly, four points are not enough: with some bad luck, three of them form a triangle and the fourth lies inside the triangle formed by the other three, and as a result, there are no four points that form a convex quadrilateral. So f(4)>4. • However, 5 points will work for any case!

12. Klein’s Cases • 1.) If the five points lie on a pentagon, then any four of them form a convex quadrilateral. • 2.) If four points form a convex quadrilateral with the remaining point in the interior, then we are done again, for the four “outer” points take on the desired shape. • 3.) Finally, if three points form a triangle and both the remaining points lie in the interior of the triangle, then one of the sides of the triangle together with the line that joins the two interior points will form a convex quadrilateral.

13. Klein’s Cases Case 1 Case 2 Case 3

14. What We Have So Far • So far the Happy Ending Problem has given us the following: • f(3) = 3, trivially. • f(4) = 5 • f(5) = 9 • f(6) = 17

15. Current Research • The value of f(N) is unknown for all N > 6 • By the result of Erdös& Szekeres (1935) it is known to be finite. • Follows from the Erdös-SzekeresConjecture: • “For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon.” • While it remains unproven, there are less precise bounds that are known • the best known upper bound when N ≥ 7 is

16. References • http://en.wikipedia.org/wiki/Discrete_geometry • http://planetmath.org/HappyEndingProblem • http://www.mathopenref.com/point.html • http://en.wikipedia.org/wiki/Convex_hull • http://www.mathsisfun.com/definitions/convex.html • http://neeldhara.com/ramblings/notes/cgt-01 • http://en.wikipedia.org/wiki/Happy_Ending_problem • http://online.sfsu.edu/meredith/301/Papers/Happy%20Ending%20Problem%20Term%20Paper%20(May%201%202011).pdf • http://cg.iit.bme.hu/~zsolnai/writings/ramsey_happy_end/ramsey_happy_end.pdf • http://sshanshans.wordpress.com/2011/12/01/the-happy-ending-problem-2/