# Milliken–Taylor theorem

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
${\displaystyle [FS(\langle a_{n}\rangle _{n=0}^{\infty })]_{<}^{k}=\left\{\left\{\sum \alpha _{1},\ldots ,\sum \alpha _{k}\right\}:\alpha _{1},\cdots ,\alpha _{k}\in {\mathcal {P}}_{f}(\mathbb {N} ){\text{ and }}\alpha _{1}<\cdots <\alpha _{k}\right\}.}$
Let ${\displaystyle [S]^{k}}$ denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition ${\displaystyle [\mathbb {N} ]^{k}=C_{1}\cup C_{2}\cup \cdots \cup C_{r}}$, there exist some ir and a sequence ${\displaystyle \langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N} }$ such that ${\displaystyle [FS(\langle a_{n}\rangle _{n=0}^{\infty })]_{<}^{k}\subset C_{i}}$.
For each ${\displaystyle \langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N} }$, 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.