Milliken–Taylor theorem explained

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

l{P}_f(N)

denote the set of finite subsets of

, and define a partial order on

l{P}_f(N)

by α<β if and only if max α

\langlea_n

\subsetN

and, let

[FS(\langlea_n

=\left\{\left\{

\sum
	t\in\alpha₁

a_t,\ldots,

\sum
	t\in\alpha_k

a_t\right\}:\alpha₁, … ,\alpha_k\inl{P}_{f(N)and\alpha}₁< … <\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

\cupC₂\cup … \cupC_r

, there exist some and a sequence

\langlea_n

\subsetN

such that

[FS(\langlea_n

\subsetC_i

For each

\langlea_n

\subsetN

, call

[FS(\langlea_n

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