Schur\u27s Theorem and Related Topics in Ramsey Theory

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\u27s theorem in Chapter 2, Schur\u27s theorem in Chapter 4, and Van DerWaerden\u27s theorem in Chapter 7), paying particular attention to Schur\u27s theorem. We discuss the origin (Chapter 5), proofs (Chapters 4 and 5), consequences (Chapter 6), and some generalizations (Chapter 8) of Schur\u27s theorem. Among generalizations we mention Rado and Szemerédi\u27s 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\u27s Last Theorem, namely that modular arithmetic does not suffice to provide a proof, and how this can be verified as a consequence of Schur\u27s theorem or by using Fourier analysis