This is an introductory course to combinatorial number theory.

Combinatorial number theory is a field of research in mathematics that has seen tremendous growth in recent years. It is a very interdisciplinary subject, since it incorporates ideas from a wide range of different areas: harmonic analysis, graph theory, number theory, ergodic theory, discrete geometry, probability theory, and even theoretical computer science. But rest assured, you don’t need any prerequisites from these areas to take this course, because we will keep things simple and develop everything we need along the way.

The main objective of this course is to learn how to use combinatorial, probabilistic, and analytic methods to solve problems in number theory. We will cover various results in Ramsey theory (such as Schur’s Theorem, van der Waerden’s Theorem, or the Erdos-Szekeres Theorem) and in additive combinatorics (such as Roth’s Theorem).