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 [http://www.math.union.edu/people/faculty/publications/taylora.html Alan D. Taylor] .

Let $mathcal\{P\}\_f(mathbb\{N\})$ denote the set of finite subsets of $mathbb\{N\}$. Given a sequence of integers $langle\; a\_n\; angle\_\{n=0\}^\{infty\}\; subset\; mathbb\{N\}$ and nowrap|"k" > 0 let :,where 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 nowrap|"i" < "r" + 1 and a sequence $langle\; x\_n\; angle\_\{n=0\}^\{infty\}\; subset\; mathbb\{N\}$ such that .

For each $langle\; a\_n\; angle\_\{n=0\}^\{infty\}\; subset\; mathbb\{N\}$, call an "MT^k set". Then, alternatively, the Milliken-Taylor theorem asserts that the collection of MT^"k" sets is partition regular for each "k".

References

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