Integral and Euclidean Ramsey theory

Ramsey theory is the study of unavoidable structure within a system. This idea is very broad, and also useful in many applications, so the theory is vast. The original theorem of Ramsey [32] states that given k, there is n such that for any graph G on n vertices, either G or its complement contain K_k as a subgraph. Statements like this can be made about any mathematical structure, but this dissertation will focus on sets of integers and on Euclidean space, both of which support a large literature within Ramsey theory. Finally, we will consider a problem in extremal combinatorics, a field that has a large intersection with Ramsey theory