Ergodic approaches to arithmetic Ramsey theory

Ramsey theory is a branch of combinatorics which seeks to find patterns in disorganized situations. One of its main achievements, Szemeredi’s theorem on arithmetic progressions, got an ergodic theoretic proof in 1977 when Furstenberg created a Correspondence Principle to encode combinatorial information about sets of integers into a dynamical system. Since then ergodic methods have been very successful in obtaining new Ramsey theoretic results, some of which still have no purely combinatorial proof.

I will survey some of the history of how ergodic theory and Ramsey theory are interconnected, leading to a recent result involving infinite sumsets.