***Distinguished Visitor Colloquium***
Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemerédi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used ergodic theory to give a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. Based on joint work with Joel Moreira, Florian Richter, and Donald Robertson, we discuss recent developments for infinite patterns, including the resolution of a conjecture of Erdos.