Ramsey’s theorem

Abstract

Ramsey theory studies, generally speaking, the following problem: Suppose that a given structure is colored using finitely many colors (equivalently, partition into finitely many pieces). Which combinatorial configurations can be found that are monochromatic, i.e. consisting of elements of the same color (equivalently, entirely contained in one of the pieces)? Ramsey’s theorem from 1930, which we will present in this chapter, can be seen as the foundational result in this area. While remarkably simple to state, it has a large number of important consequences and applications. Many of these applications were studied by Erdős and Rado in the 1950s, who “rediscovered” Ramsey’s theorem and recognized it importance. Attempts to generalize Ramsey’s theorem in different contexts and directions have been one of the main driving forces in Ramsey theory.