Milliken–Taylor theorem
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, and define a partial order on by α<β if and only if max αsequence of integers and, let
Let 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.
