Ramsey theory
From Academic Kids

Ramsey theory, named for 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? An oftquoted slogan for the subject is "complete disorder is impossible".
Suppose, for example, that we know that n pigeons have been housed in m pigeonholes. How big must n be before we can be sure that at least one pigeonhole houses at least two pigeons? The answer is the pigeonhole principle: if n > m, then at least one pigeonhole will have at least two pigeons in it. Ramsey's theorem generalizes this principle as explained below.
A typical result in Ramsey theory starts with some mathematical structure, which 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?
For example, consider a complete graph of order n, that is, there are n vertices (dots) and each vertex is connected to every other vertex by an edge (a line). A complete graph of order 3 is called a triangle. Now color 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 either three people who are all mutual acquaintances (each one knows the other two) or mutual strangers (each one does not know either of the other two).
This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n_{1},...,n_{c}, there is a number, R(n_{1},...,n_{c};c), such that if the edges of a complete graph of order R(n_{1},...,n_{c};c) are colored with c different colors, then for some i between 1 and c, it must contain a complete subgraph of order n_{i} whose edges are all color i. The special case above has c = 2 and n_{1} = n_{2} = 3.
Three other 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 the elements of an arithmetic progression of V numbers are colored with c different colors, then it must contain an arithmetic progression of length n whose elements are all the same color.
 HalesJewett theorem: For any given n and c, there is a number H such that if the cells of a Hdimensional n×n×n×...×n cube are colored with c colors, there must be one row, column, etc. of length n all of whose cells are the same color. That is, if you play on a board with sufficiently many dimensions, then multiplayer ninarow tictactoe cannot end in a draw, no matter how large n is, and no matter how many people are playing.
A variant of 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 colored with c different colors, then there must be a pair of integers x, y such that x, y, and x+y are all the same color. Many generalizations of this theorem exist, including Rado's Theorem, RadoFolkmanSanders' theorem, and Hindman's theorem. A classic reference for these and many other results in Ramsey theory is [1].
Results in Ramsey theory typically have two notable characteristics. Firstly, they are often nonconstructive; they may show that some structure exists, but they give no recipe for how to actually find this structure (other than brute force search); for instance, the pigeonhole principle is of this form. Secondly, while Ramsey theory results do say that sufficiently large objects must necessarily contain a given structure, often the proof of these results requires these objects to be enormously large  bounds which grow exponentially, or even as fast as the Ackermann function are not uncommon. In many cases these bounds are artefacts of the proof, and it is not known whether they can be substantially improved.
References
 R. Graham, B. Rothschild, J.H. Spencer, Ramsey Theory, John Wiley and Sons, NY (1980).
 B. Landman and A. Robertson, Ramsey Theory on the Integers, Student Mathematical Library Vol. 24, AMS, (2004)