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An Introduction to the Happy Ending Problem and the Erdos–Szekeres Conjecture by Joseph Bedich.pdf (480.75 KB)
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Abstract Header
An Introduction to the Happy Ending Problem and the Erdős–Szekeres Conjecture
Author Info
Bedich, Joseph Matthew
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu152405396768852
Abstract Details
Year and Degree
2018, Master of Mathematical Sciences, Ohio State University, Mathematics.
Abstract
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)
= 2
n-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.
Committee
Matthew Kahle (Advisor)
Herb Clemens (Committee Member)
Subject Headings
Mathematics
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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)
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Document number:
osu152405396768852
Download Count:
2,575
Copyright Info
© 2018, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.