Ramsey theory

Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order n; that is, there are n vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must n be in order to ensure that there is either a blue triangle or a red triangle? It turns out that the answer is 6. See the article on Ramsey's theorem for a rigorous proof. Another way to express this result is as follows: at any party with at least six people, there are three people who are all either mutual acquaintances (each one knows the other two) or mutual strangers (none of them knows either of the other two). See theorem on friends and strangers. This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n1,...,nc, there is a number, R(n1,...,nc), such that if the edges of a complete graph of order R(n1,...,nc) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all colour i. The special case above has c = 2 and n1 = n2 = 3. Two key theorems of Ramsey theory are: Van der Waerden's theorem: For any given c and n, there is a number V, such that if V consecutive numbers are coloured with c different colours, then it must contain an arithmetic progression of length n whose elements are all the same colour.