Polynomial extensions of the Milliken-Taylor Theorem
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- by Vitaly Bergelson, Neil Hindman and Kendall Williams PDF
- Trans. Amer. Math. Soc. 366 (2014), 5727-5748 Request permission
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 $\beta {\mathcal F}$.References
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Additional Information
- Vitaly Bergelson
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
- MR Author ID: 35155
- Email: vitaly@math.ohio-state.edu
- Neil Hindman
- Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
- MR Author ID: 86085
- 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
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3256182