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# Ramsey Number

The Ramsey number gives the solution to the party problem, which asks the minimum number of guests that must be invited so that at least will know each other or at least will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices such that all undirected simple graphs of order contain a clique of order or an independent set of order . Ramsey's theorem states that such a number exists for all and .

By symmetry, it is true that

 (1)

It also must be true that

 (2)

A generalized Ramsey number is written

 (3)

and is the smallest integer such that, no matter how each -element subset of an -element set is colored with colors, there exists an such that there is a subset of size , all of whose -element subsets are color . The usual Ramsey numbers are then equivalent to .

Bounds are given by

 (4)

and

 (5)

(Chung and Grinstead 1983). Erdős proved that for diagonal Ramsey numbers ,

 (6)

This result was subsequently improved by a factor of 2 by Spencer (1975). was known since 1980 to be bounded from above by , and Griggs (1983) showed that was an acceptable limit. J.-H. Kim (Cipra 1995) subsequently bounded by a similar expression from below, so

 (7)

Burr (1983) gives Ramsey numbers for all 113 graphs with no more than 6 graph edges and no isolated points.

A summary of known results up to 1983 for is given in Chung and Grinstead (1983). Radziszowski (2004) maintains an up-to-date list of the best current bounds. Results from Tables I and II of Radziszowski (2004) are reproduced below in a slightly less cramped format than in the original. Known bounds for generalized Ramsey numbers (multicolor graph numbers), hypergraph Ramsey numbers, and many other types of Ramsey numbers may be found in Radziszowski (2004). In the absence of a published upper bound, the theorem of Erdős-Szekeres stating that is used to provide one.

Clique, Clique Number, Complete Graph, Extremal Graph, Independence Number, Independent Set, Irredundant Ramsey Number, Ramsey's Theorem, Ramsey Theory, Schur Number

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