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.

Statement

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

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

$$\chi (H)\le 1+\frac{\ln m}{-\ln (1-\alpha (H)/m)}$$

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).

See also

• Ramsey theory