Ramsey's Theorem

Ramsey's theorem is a generalization of Dilworth's lemma which states for each pair of positive integers k and l there exists an integer R(k,l) (known as the Ramsey number) such that any graph with R(k,l) nodes contains a clique with at least k nodes or an independent set with at least l nodes.

Another statement of the theorem is that for integers k,l>=2, there exists a least positive integer R(k,l) such that no matter how the complete graph K_(R(k,l)) is two-colored, it will contain a green subgraph K_k or a red subgraph K_l.

A third statement of the theorem states that for all k in N, there exists an l in N such that any complete digraph on l graph vertices contains a complete vertex-transitive subgraph of k graph vertices.

For example, R(3,3)=6 and R(4,4)=18, but R(5,5) are only known to lie in the ranges 43<=R(5,5)<=49 and 102<=R(6,6)<=165.

It is true that


if k,l>=3.

See also

Dilworth's Lemma, Dirichlet's Box Principle, Extremal Graph Theory, Graph Coloring, Natural Independence Phenomenon, Party Problem, Ramsey Number, Ramsey Theory

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Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 33-34, 2003.Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990.Mileti, J. "Ramsey's Theorem.", J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669-675, 1983.

Referenced on Wolfram|Alpha

Ramsey's Theorem

Cite this as:

Weisstein, Eric W. "Ramsey's Theorem." From MathWorld--A Wolfram Web Resource.

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