Arithmetic Ramsey Theory with a Focus on Exponential Patterns

To get some feel for the subject, the mini-course shall start gently with an overview of some basic topics such as van der Waerden’s classical theorem: that every finite colouring of the integers contains arbitrarily long monochromatic arithmetic progressions. We shall then move quickly to the more recent non-linear theory; for example, a theorem of Furstenberg and  Sárkōsy sates that every finite colouring  of the  integers contains a monochromatic pair x, y that differ in a perfect square. In the main segment of this mini-course, we shall discuss the results and techniques in the new area of exponential patterns in arithmetic Ramsey Theory.  For example, we shall prove that every finite colouring of the integers  contains  a monochromatic triple  of the form {a, b, ab }, where a, b > 1. From here we shall discuss many further  results  and open problems.  The mini-course will take  a combinatorial perspective throughout and should be accessible to all.