Publication Date

5-2013

Type of Culminating Activity

Thesis - Boise State University Access Only

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Andrés E. Caicedo, Ph.D.

Abstract

Ramsey theory is a rich field of study and an active area of research. The theory can best be described as a combination of set theory and combinatorics; however, the arguments to prove some of its results vary across many fields. In this thesis, we survey the field of Ramsey theory highlighting three of its main theorems (Ramsey's theorem in Chapter 2, Schur's theorem in Chapter 4, and Van DerWaerden's theorem in Chapter 7), paying particular attention to Schur's theorem.

We discuss the origin (Chapter 5), proofs (Chapters 4 and 5), consequences (Chapter 6), and some generalizations (Chapter 8) of Schur's theorem. Among generalizations we mention Rado and Szemerédi's theorems. Special attention is also paid to upper and lower bounds for Schur numbers.

Along the way, we take a couple detours, going into areas of mathematics or history that are relevant to what we are studying. In particular Chapter 3 includes a biography of Issai Schur, and Chapter 6 discusses results related to Fermat's Last Theorem, namely that modular arithmetic does not suffice to provide a proof, and how this can be verified as a consequence of Schur's theorem or by using Fourier analysis.

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