Definition:Ramsey Number

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 $$n_1, \ldots, n_c$$.