Symmetric hypergraph theorem

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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

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