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An Introduction to the Happy Ending Problem and the Erdős–Szekeres Conjecture

Bedich, Joseph Matthew

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2018, Master of Mathematical Sciences, Ohio State University, Mathematics.
In 1932, Esther Klein made the observation that among any 5 points in general position in the plane, one can always find 4 points that form a convex quadrilateral. This problem came to be known as the Happy Ending Problem. She then posed the following generalization of her observation: how many points in the plane are needed to guarantee a convex n-gon, if such a number of points exists at all? Two of the mathematicians Klein was working with at the time, Paul Erdős and George Szekeres, published their progress on Klein’s generalized problem in 1935. Their paper proved that for any natural number n ≥ 3, there exists a minimal number ES(n) such that any configuration of ES(n) points in general position in the plane is guaranteed to contain a convex n-gon. They also attempted to find a formula for ES(n), and while they did not actually prove that ES(n) could be written as a function of n, they did conjecture that ES(n) = 2n-2 + 1. This has come to be known as the Erdős–Szekeres Conjecture. Despite the work of many mathematicians since, this conjecture still remains unproven. This paper details the history of the Happy Ending Problem and the Erdős–Szekeres Conjecture, provides proofs for values of ES(n) for some specific n, and discusses the progress that has been made toward proving the Erdős–Szekeres Conjecture.
Matthew Kahle (Advisor)
Herb Clemens (Committee Member)

Recommended Citations

Citations

  • Bedich, J. M. (2018). An Introduction to the Happy Ending Problem and the Erdős–Szekeres Conjecture [Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu152405396768852

    APA Style (7th edition)

  • Bedich, Joseph. An Introduction to the Happy Ending Problem and the Erdős–Szekeres Conjecture. 2018. Ohio State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu152405396768852.

    MLA Style (8th edition)

  • Bedich, Joseph. "An Introduction to the Happy Ending Problem and the Erdős–Szekeres Conjecture." Master's thesis, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu152405396768852

    Chicago Manual of Style (17th edition)