Milliken–Taylor theorem

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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 α<min β. Given a sequence of integers and k > 0, let

Let denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition , there exist some ir 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.