---
tag: G5TWXV565S
kind: concept
name: Ramsey theory
msc: [05D10, 05C55]
sources:
  - url: https://en.wikipedia.org/wiki/Ramsey_theory
same_as:
  wikidata: Q1336170
  wikipedia: 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|00TQV3F5S4]] on monochromatic cliques in coloured complete graphs, with the [[Ramsey numbers|0K9QVPJYJF]] as its quantitative side.
In the integers it includes [[Schur's theorem|V6XJQXZ7E8]], [[van der Waerden's theorem|V96FRZ2ME1]] and [[Rado's theorem|BZSGRCTFH4]], and the abstract [[Hales–Jewett theorem|14SGZ5HA4Y]] contains many of these.
Density versions, such as [[Szemerédi's theorem|B23B0QEMN5]], link it to [[additive combinatorics|3NM362A2JG]] and ergodic theory.

## Explanations

### L1 human

Ramsey theory says that complete disorder is impossible.
If you colour the objects of a big enough structure with a few colours, some pattern always appears in a single colour.
The [[pigeonhole principle|0HGVCWQGY3]] is the simplest case: with $n$ colours and $n + 1$ objects, two objects get the same colour.
A famous example: among any six people, there are three who all know each other or three who are all strangers to each other.
The hard questions are how big "big enough" must be, and these numbers are often unknown even for small cases.