Milliken–Taylor theorem

Template:Short description Template:Technical 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 ${𝒫}_f(\mathbb{N})$ denote the set of finite subsets of $\mathbb{N}$, and define a partial order on ${𝒫}_f(\mathbb{N})$ by α<β if and only if max α<min β. Given a sequence of integers $⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}\subset \mathbb{N}$ and Template:Nowrap, let

$[FS(⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}){]}_{<}^k=\{\{\sum _{t\in {\alpha }_1}{a}_t,\dots ,\sum _{t\in {\alpha }_k}{a}_t\}:{\alpha }_1,\cdots ,{\alpha }_k\in {𝒫}_f(\mathbb{N})\text{ and }{\alpha }_1<\cdots <{\alpha }_k\}.$

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 Template:Nowrap and a sequence $⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}\subset \mathbb{N}$ such that $[FS(⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}){]}_{<}^k\subset C_i$.

For each $⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}\subset \mathbb{N}$, call $[FS(⟨a_n{⟩}_{n=0}^{\mathrm{\infty }}){]}_{<}^k$ 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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