On partition regular systems
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An equation or system of equations is called ``partition regular in a set S" if and only if for any finite colouring of S a solution to the system is guaranteed to be contained in some colour class. This thesis is a survey of partition regular systems, starting with early results in arithmetic Ramsey theory, including Hilbert's cube lemma, Schur's theorem, and van der Waerden's theorem. A proof is given of Rado's characterization of all finite partition regular systems of homogeneous linear equations, and results concerning infinite and nonlinear partition regular systems are also proved. Several tools, including linear algebra and topology, are used in the proofs in this thesis.