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

${\displaystyle [FS(\langle a_{n}\rangle _{n=0}^{\infty })]_{<}^{k}=\left\{\left\{\sum _{t\in \alpha _{1}}a_{t},\ldots ,\sum _{t\in \alpha _{k}}a_{t}\right\}:\alpha _{1},\cdots ,\alpha _{k}\in {\mathcal {P}}_{f}(\mathbb {N} ){\text{ and }}\alpha _{1}<\cdots <\alpha _{k}\right\}.}$

Let ${\displaystyle [S]^{k}}$ 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 i ≤ r 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 MT^k set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT^k sets is partition regular for each k.

• Milliken, Keith R. (1975), "Ramsey's theorem with sums or unions", Journal of Combinatorial Theory, Series A, 18 (3): 276–290, doi:10.1016/0097-3165(75)90039-4, MR 0373906.
• Taylor, Alan D. (1976), "A canonical partition relation for finite subsets of ω", Journal of Combinatorial Theory, Series A, 21 (2): 137–146, doi:10.1016/0097-3165(76)90058-3, MR 0424571.