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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 i ≤ r 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.
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