Old questions in additive combinatorics and Ramsey theory ask what infinite patterns are unavoidable in sets of integer numbers with positive density (i.e. that account for a positive proportion of all numbers), but until recently, there were no positive answers. In the last few years a new technique was developed to address such questions, making use of ergodic theory and dynamical systems, which were previously restricted to handling finite patterns.

In this talk I will briefly survey the history of the subject, and explain the connection to ergodic theory. Then I will describe how we used this technique to answer a question of Paul Erdos regarding infinite sumsets. The talk will be based on joint work with Bryna Kra, Florian Richter and Donald Robertson.