Definition:Ramsey Number

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 $R \left({n_1, \ldots, n_c}\right)$ such that:
 * if the edges of a complete graph of order $R \left({n_1, \ldots, n_c}\right)$ are colored with $c$ different colours, 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 $R \left({n_1, \ldots, n_c}\right)$ is called the Ramsey number for $n_1, \ldots, n_c$.