# 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 ${\displaystyle G}$ acting on a set ${\displaystyle S}$ is called transitive if given any two elements ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle S}$, there exists an element ${\displaystyle f}$ of ${\displaystyle G}$ such that ${\displaystyle f(x)=y}$. A graph (or hypergraph) is called symmetric if it's automorphism group is transitive.

Theorem. Let ${\displaystyle H=(S,E)}$ be a symmetric hypergraph. Let ${\displaystyle m=|S|}$, and let ${\displaystyle \chi (H)}$ denote the chromatic number of ${\displaystyle H}$, and let ${\displaystyle \alpha (H)}$ denote the independence number of ${\displaystyle H}$. Then

${\displaystyle \chi (H)<1+{\frac {m}{\alpha (H)}}\ln m.}$

## 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).