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 ( N ) denote the set of finite subsets of N , and define a partial order on P f ( N ) by α<β if and only if max α ⟨ a n ⟩ n = 0 ∞ ⊂ N and k > 0, let
[ F S ( ⟨ a n ⟩ n = 0 ∞ ) ] < k = { { ∑ α 1 , … , ∑ α k } : α 1 , ⋯ , α k ∈ P f ( N ) and α 1 < ⋯ < α k } . Let [ S ] k denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition [ N ] k = C 1 ∪ C 2 ∪ ⋯ ∪ C r , there exist some i ≤ r and a sequence ⟨ a n ⟩ n = 0 ∞ ⊂ N such that [ F S ( ⟨ a n ⟩ n = 0 ∞ ) ] < k ⊂ C i .
For each ⟨ a n ⟩ n = 0 ∞ ⊂ N , call [ F S ( ⟨ a n ⟩ n = 0 ∞ ) ] < k an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.
