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

Another statement of the theorem is that for integers , there exists a least positive integer such that no matter how the complete
graph
is two-colored, it will contain a green subgraph or a red subgraph .