The mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large. Ramsey theory is named after Frank Plumpton Ramsey, who did seminal work in this area before his untimely death at age 26 in 1930. The theory was subsequently developed extensively by Erdős.
The classical problem in Ramsey theory is the party problem, which asks the minimum number of guests 
that must be invited so that at least 
will know each other (i.e., there exists a clique
of order 
)
or at least 
will not know each other (i.e., there exists an independent
set of order 
.
Here, 
is called a Ramsey number.
A typical result in Ramsey theory states that if some mathematical object is partitioned into finitely many parts, then one of the parts must contain a subobject of an interesting
kind. For example, it is known that if 
is large enough and 
is an 
-dimensional vector space over
the field of integers (mod 
), then however 
is partitioned into 
pieces, one of the pieces contains an affine subspace of dimension
.
