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thesis 424.pdf (443.13 KB)
ETD Abstract Container
Abstract Header
On Four-Free Sets
Author Info
Gomez-Leos, Enrique
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1587658105848612
Abstract Details
Year and Degree
2020, Master of Mathematical Sciences, Ohio State University, Mathematical Sciences.
Abstract
In this thesis we provide an exposition of Szemere'di’s theorem for the specific case of κ=4 which precedes Szemer\'{e}di's more famous result. The proof is elementary. The case of $k=3$, better known as Roth's theorem is proved as a corollary. Chapter 2 will introduce the lemmas required to prove Szemer\'{e}di's theorem for 4-terms, they are Lemmas~$BCDE$ and $(H_{0},\dots,H_{k})$. The chapter will conclude with a proof of the theorem. In chapter 3 we will prove Lemma $(H_{0},\dots,H_{k})$ along with 3 other necessary lemmas. These are the so called Simple Lemma, Lemma $p(\delta,\ell)$, and Lemma $|G^*|$ respectively. Afterwards we discuss how, with some modification to Lemma $|G^*|$, we may formulate and prove Roth's theorem as a corollary. Finally, chapter 4 will be entirely on the proof of Lemma~$BCDE$. Being the most involved in Szemeredi's construction, it will use several arguments made throughout the paper in addition to new ideas.
Committee
John Johnson (Other)
Vitaly Bergelson (Advisor)
Alexander Leibman (Committee Member)
Pages
64 p.
Subject Headings
Mathematics
Keywords
combinatorics
;
Ramsey theory
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Citations
Gomez-Leos, E. (2020).
On Four-Free Sets
[Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587658105848612
APA Style (7th edition)
Gomez-Leos, Enrique.
On Four-Free Sets.
2020. Ohio State University, Master's thesis.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1587658105848612.
MLA Style (8th edition)
Gomez-Leos, Enrique. "On Four-Free Sets." Master's thesis, Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587658105848612
Chicago Manual of Style (17th edition)
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Document number:
osu1587658105848612
Download Count:
37
Copyright Info
© 2020, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.