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.

(

denote the set of finite subsets of

{\displaystyle \mathbb {N} }

, and define a partial order on

(

by α<β if and only if max α

Given a sequence of integers

denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition

, there exist some i ≤ r and a sequence

{\displaystyle [FS(\langle a_{n}\rangle _{n=0}^{\infty })]_{<}^{k}\subset C_{i}}

, call

{\displaystyle [FS(\langle a_{n}\rangle _{n=0}^{\infty })]_{<}^{k}}

an MTk set.

Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.

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