Ramsey theory

Summary

Ramsey theory studies the statements that every large enough structure, divided into finitely many parts, has a part that contains a given regular substructure. Its founding result is Ramsey's theorem on monochromatic cliques in coloured complete graphs, with the Ramsey numbers as its quantitative side. In the integers it includes Schur's theorem, van der Waerden's theorem and Rado's theorem, and the abstract Hales–Jewett theorem contains many of these. Density versions, such as Szemerédi's theorem, link it to additive combinatorics and ergodic theory.