Polynomial extensions of the Milliken-Taylor Theorem

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 $\beta {\mathcal F}$.