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

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

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).