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 $$\mathcal{P}_f(\mathbb{N})$$ denote the set of finite subsets of $$\mathbb{N}$$, and define a partial order on $$\mathcal{P}_f(\mathbb{N})$$ by &alpha;<&beta; if and only if max &alpha; 0, let
 * $$[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< = \left \{ \left \{ \sum_{t\in \alpha_1}a_t, \ldots, \sum_{t\in \alpha_k}a_t \right \}: \alpha_1 ,\cdots , \alpha_k \in \mathcal{P}_f(\mathbb{N})\text{ and }\alpha_1 <\cdots < \alpha_k \right \}.$$

Let $$[S]^k$$ denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition $$[\mathbb{N}]^k=C_1 \cup C_2 \cup \cdots \cup C_r$$, there exist some i &le; r and a sequence $$\langle a_n \rangle_{n=0}^{\infty} \subset \mathbb{N}$$ such that $$[FS(\langle a_n \rangle_{n=0}^{\infty})]^k_< \subset C_i$$.

For each $$\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}$$, call $$[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< $$ an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.