Ramsey's theorem

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph (that is, a simple graph in which an edge connects every pair of vertices), one will find monochromatic complete subgraphs. For 2 colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.