# Symmetric hypergraph theorem

The **Symmetric hypergraph theorem** is a theorem in combinatorics that puts an upper bound on the chromatic number of a graph (or hypergraph in general). The original reference for this paper is unknown at the moment, and has been called folklore.^{[1]}

## Statement

A group acting on a set is called *transitive* if given any two elements and in , there exists an element of such that . A graph (or hypergraph) is called *symmetric* if it's automorphism group is transitive.

**Theorem.** Let be a symmetric hypergraph. Let , and let denote the chromatic number of , and let denote the independence number of . Then

## Applications

This theorem has applications to Ramsey theory, specifically graph Ramsey theory. Using this theorem, a relationship between the graph Ramsey numbers and the extremal numbers can be shown (see Graham-Rothschild-Spencer for the details).

## See also

## Notes

- ↑ R. Graham, B. Rothschild, J. Spencer. Ramsey Theory. 2nd ed., Wiley, New-York, 1990.