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 denote the set of finite subsets of , and define a partial order on by α<β if and only if max α<min β. Given a sequence of integers and k > 0, let

Let denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition , there exist some ir and a sequence such that .

For each , call an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.


  • Milliken, Keith R. (1975), "Ramsey's theorem with sums or unions", Journal of Combinatorial Theory, Series A, 18: 276–290, doi:10.1016/0097-3165(75)90039-4, MR 0373906.
  • Taylor, Alan D. (1976), "A canonical partition relation for finite subsets of ω", Journal of Combinatorial Theory, Series A, 21 (2): 137–146, doi:10.1016/0097-3165(76)90058-3, MR 0424571.