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 ${\displaystyle {\mathcal {P}}_{f}(\mathbb {N} )}$ denote the set of finite subsets of ${\displaystyle \mathbb {N} }$, and define a partial order on ${\displaystyle {\mathcal {P}}_{f}(\mathbb {N} )}$ by α<β if and only if max α<min β. Given a sequence of integers ${\displaystyle \langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N} }$ and k > 0, let

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