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 \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 $[S]^{k}$ 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: 276–290, MR 0373906, doi:10.1016/0097-3165(75)90039-4 .
Taylor, Alan D. (1976), "A canonical partition relation for finite subsets of ω", Journal of Combinatorial Theory, Series A, 21 (2): 137–146, MR 0424571, doi:10.1016/0097-3165(76)90058-3 .

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