Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"
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 every edge 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 (each one does not know 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:
A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there is a number N such that if the numbers 1, 2, ..., N are coloured with c different colours, then there must be a pair of integers x, y such that x, y, and x+y are all the same colour. Many generalizations of this theorem exist, including Rado's theorem, Rado-Folkman-Sanders theorem, Hindman's theorem, and the Milliken-Taylor theorem. A classic reference for these and many other results in Ramsey theory is Graham, Rothschild and Spencer.
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