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| In [[mathematics]], the '''Milliken–Taylor theorem''' in [[combinatorics]] is a generalization of both [[Ramsey's theorem]] and [[IP set|Hindman's theorem]]. It is named after Keith Milliken and [http://www.math.union.edu/people/faculty/publications/taylora.html Alan D. Taylor].
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| Let <math>\mathcal{P}_f(\mathbb{N})</math> denote the set of finite subsets of <math>\mathbb{N}</math>, and define a partial order on <math>\mathcal{P}_f(\mathbb{N})</math> by α<β [[if and only if]] max α<min β. Given a sequence of integers <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math> and {{nowrap|''k'' > 0}}, let
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| :<math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< = \left \{ \left \{ \sum \alpha_1, \ldots , \sum \alpha_k \right \}: \alpha_1 ,\cdots , \alpha_k \in \mathcal{P}_f(\mathbb{N})\text{ and }\alpha_1 <\cdots < \alpha_k \right \}.</math>
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| Let <math>[S]^k</math> denote the ''k''-element subsets of a set ''S''. The Milliken–Taylor theorem says that for any finite partition <math>[\mathbb{N}]^k=C_1 \cup C_2 \cup \cdots \cup C_r</math>, there exist some {{nowrap|''i'' ≤ ''r''}} and a sequence <math>\langle a_n \rangle_{n=0}^{\infty} \subset \mathbb{N}</math> such that <math>[FS(\langle a_n \rangle_{n=0}^{\infty})]^k_< \subset C_i</math>.
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| For each <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math>, call <math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< </math> an ''MT<sup>k</sup> set''. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT<sup>''k''</sup> sets is [[partition regular]] for each ''k''.
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| ==References==
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| #K. Milliken, Ramsey's Theorem with sums or unions, ''J. Comb. Theory (Series A)'' '''18''' (1975), 276–290 [http://dx.doi.org/10.1016/0097-3165(75)90039-4]
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| #A. Taylor, A canonical partition relation for finite subsets of ω, ''J. Comb. Theory (Series A)'' '''21''' (1976), 137–146 [http://dx.doi.org/10.1016/0097-3165(76)90058-3]
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| {{DEFAULTSORT:Milliken-Taylor theorem}}
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| [[Category:Ramsey theory]]
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| [[Category:Theorems in discrete mathematics]]
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| {{combin-stub}}
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