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 α<β if and only if max α<min β. Given a sequence of integers $\langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N}$ and k > 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 ≤ 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 MT^k set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT^k sets is partition regular for each k.

References

• 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.