Definition:Ramsey Number

This page has been identified as a candidate for refactoring of medium complexity. In particular: so as to define this without restating the premises of Ramsey's Theorem from scratch -- maybe transclude it if need be. In any way, find a more streamlined way to present this. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Definition

Ramsey's Theorem states that in any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs.

More precisely, for any given number of colors $c$, and any given integers $n_1, \ldots, n_c$, there is a number $\map R {n_1, \ldots, n_c}$ such that:

if the edges of a complete graph of order $\map R {n_1, \ldots, n_c}$ are colored with $c$ different colors, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$.

This number $\map R {n_1, \ldots, n_c}$ is called the Ramsey number for $n_1, \ldots, n_c$.

Also see

• Results about Ramsey numbers can be found here.

Source of Name

This entry was named for Frank Plumpton Ramsey.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Ramsey's theorem (F.P. Ramsey, 1928)