Milliken-Taylor theorem
Encyclopedia
In mathematics
, the Milliken–Taylor theorem in combinatorics
is a generalization of both Ramsey's theorem
and Hindman's theorem
. It is named after Keith Milliken and Alan D. Taylor.
Let
denote the set of finite subsets of 
. Given a sequence of integers 
and let
,
where
if and only if
max α denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition 
, there exist some and a sequence 
such that 
.
For each
, call 
an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Milliken–Taylor theorem in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
is a generalization of both Ramsey's theorem
Ramsey's theorem
In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs...
and Hindman's theorem
IP set
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D.The set of all finite sums over D is often...
. It is named after Keith Milliken and Alan D. Taylor.
Let
denote the set of finite subsets of . Given a sequence of integers and let,where
if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
max α
, there exist some and a sequence such that .For each
, call an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.