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Milliken taylor theorem&o=10616

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 mathcal{P}_f(mathbb{N}) denote the set of finite subsets of mathbb{N}. Given a sequence of integers langle a_n rangle_{n=0}^{infty} subset mathbb{N} and let

[FS(langle a_n rangle_{n=0}^{infty})]^k_< = left { left { sum_{t in alpha_1}x_t, ... , sum_{t in alpha_k}x_t right }: alpha_1 <...< alpha_k in mathcal{P}_f(mathbb{N}) right },
where alpha < beta in mathcal{P}_f(mathbb{N}) if and only if maxα[S]^k denote the k-element subsets of a set S. The Milliken-Taylor theorem says that for any finite partition [mathbb{N}]^k=C_1 cup C_2 cup ... cup C_r, there exist some and a sequence langle x_n rangle_{n=0}^{infty} subset mathbb{N} such that [FS(langle a_n rangle_{n=0}^{infty})]^k_< subset C_i.

For each langle a_n rangle_{n=0}^{infty} subset mathbb{N}, call [FS(langle a_n rangle_{n=0}^{infty})]^k_< an MTk set. Then, alternatively, the Milliken-Taylor theorem asserts that the collection of MTk sets is partition regular for each k.

References

  1. K. Milliken, Ramsey's Theorem with sums or unions, J. Comb. Theory (Series A) 18 (1975), 276-290
  2. A. Taylor, A canonical partition relation for finite subsets of ω, J. Comb. Theory (Series A) 21 (1976), 137-146

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Last updated on Monday October 06, 2008 at 05:50:46 PDT (GMT -0700)
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