Abstract: Consider six points in the plane and connect (or not) each pair of those at random. Many graphs can be obtained that way but all of them share the following property: there are three points where the pairs are all connected or never connected. This simple combinatorial result is at the base of what is called Ramsey theory, a theory that describes the appearance of unavoidable patterns and nowadays touches areas of mathematics that are seemingly very far from combinatorics. The purpose of this talk will be to present some of those connections taken from metric geometry and topological group theory.