Arithmetic Ramsey Theory

Arithmetic Ramsey Theory is a branch of combinatorics concerned with the abundance of solutions to certain arithmetic problems on the natural numbers. A typical example of such a problem is determining whether a given configuration is ubiquitous enough that one is forced to find instances of it in each piece of every finite partition of the naturals. In the course, we will focus on how ultrafilters and model-theoretical methods can be used to give short and informative proofs of many fundamental results in this area, mostly related to Diophantine equations, including the theorems of Ramsey, Schur, Van der Waerden, Rado, Hindman, and various generalizations. If time allows, we will also present some results related to certain fundamental open problems in the area.