A generalization of Ramsey theory to mathematical objects in which one would not normally expect structure to be found. For example, there exists a graph with very few triangles (more precisely, a graph which can always be constructed so that there is no "cycle" of triangles which are all distinct and T_i meets T_(i + 1) in at least one vertex) and such that however it is colored with r colors, one of the colors contains a triangle. The usual proof of Ramsey's theorem gives no insight on how to prove such a result.

extremal graph theory | Ramsey's theorem | Ramsey theory