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Polynomial extensions of the Milliken-Taylor Theorem


Authors: Vitaly Bergelson, Neil Hindman and Kendall Williams
Journal: Trans. Amer. Math. Soc. 366 (2014), 5727-5748
MSC (2010): Primary 03E05, 05D10
DOI: https://doi.org/10.1090/S0002-9947-2014-05958-8
Published electronically: June 16, 2014
MathSciNet review: 3256182
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Abstract: Milliken-Taylor systems are some of the most general infinitary configurations that are known to be partition regular. These are sets of the form $ MT(\langle a_i\rangle _{i=1}^m,\langle x_n\rangle _{n=1}^\infty )= \{\sum _{i=1}^m a_i\sum _{t\in F_i}\,x_t:F_1,F_2,\ldots , F_m$ are increasing finite nonempty subsets of $ \mathbb{N}\}$, where $ a_1,a_2,\ldots ,a_m\in \mathbb{Z}$ with $ a_m>0$ and $ \langle x_n\rangle _{n=1}^\infty $ is a sequence in $ \mathbb{N}$. That is, if $ p(y_1,y_2,\ldots ,y_m)=\sum _{i=1}^m a_iy_i$ is a given linear polynomial and a finite coloring of $ \mathbb{N}$ is given, one gets a sequence $ \langle x_n\rangle _{n=1}^\infty $ such that all sums of the form $ p(\sum _{t\in F_1}x_t,\ldots ,\sum _{t\in F_m}x_t)$ are monochromatic. In this paper we extend these systems to images of very general extended polynomials. We work with the Stone-Čech compactification $ \beta {\mathcal F}$ of the discrete space $ {\mathcal F}$ of finite subsets of $ \mathbb{N}$, whose points we take to be the ultrafilters on $ {\mathcal F}$. We utilize a simply stated result about the tensor products of ultrafilters and the algebraic structure of $ \beta {\mathcal F}$.


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Additional Information

Vitaly Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: vitaly@math.ohio-state.edu

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Email: nhindman@aol.com

Kendall Williams
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Address at time of publication: Department of Mathematical Sciences, United States Military Academy, West Point, New York 10996
Email: kendallwilliams1983@yahoo.com, Kendall.Williams@usma.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05958-8
Received by editor(s): September 19, 2011
Received by editor(s) in revised form: September 10, 2012
Published electronically: June 16, 2014
Additional Notes: The first two authors acknowledge support received from the National Science Foundation via Grants DMS-0901106 and DMS-0852512 respectively.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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