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 $\mathcal_f(\mathbb)$ denote the set of finite subsets of $\mathbb$, and define a partial order on $\mathcal_f(\mathbb)$ by α<β if and only if max α\langle a_n \rangle_^\infty \subset \mathbb and , let
: S(\langle a_n \rangle_^\infty)k_< = \left \.
Let k denote the ''k''-element subsets of a set ''S''. The Milliken–Taylor theorem says that for any finite partition $$mathbbk=C_1 \cup C_2 \cup \cdots \cup C_r, there exist some and a sequence $\langle a_n \rangle_^ \subset \mathbb$ such that S(\langle a_n \rangle_^)k_< \subset C_i.

For each $\langle a_n \rangle_^\infty \subset \mathbb$, call S(\langle a_n \rangle_^\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

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Ramsey theory
Theorems in discrete mathematics

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