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 $G$ 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

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

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).

See also

• Ramsey theory

Notes

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