SEPARATING MILLIKEN–TAYLOR SYSTEMS WITH NEGATIVE ENTRIES

Abstract

Given a finite sequence $\bm{a}=\langle a_i\rangle_{i=1}^n$ in $\mathbb{N}$ and a sequence $\langle x_t\rangle_{t=1}^\infty$ in $\mathbb{N}$, the Milliken–Taylor system generated by $\bm{a}$ and $\langle x_t\rangle_{t=1}^\infty$ is

\begin{multline*} \qquad \mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)=\biggl\{\sum_{i=1}^na_i\cdot\sum_{t\in F_i}x_t:F_1,F_2,\dots,F_n\text{ are finite non-empty} \\[-8pt] \text{subsets of $\mathbb{N}$ with }\max F_i\lt\min F_{i+1}\text{ for }i\ltn\biggr\}.\qquad \end{multline*}

It is known that Milliken–Taylor systems are partition regular but not consistent. More precisely, if $\bm{a}$ and $\bm{b}$ are finite sequences in $\mathbb{N}$, then, except in trivial cases, there is a partition of $\mathbb{N}$ into two cells, neither of which contains $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ for any sequences $\langle x_t\rangle_{t=1}^\infty$ and $\langle y_t\rangle_{t=1}^\infty$.

Our aim in this paper is to extend the above result to allow negative entries in $\bm{a}$ and $\bm{b}$. We do so with a proof which is significantly shorter and simpler than the original proof which applied only to positive coefficients. We also derive some results concerning the existence of solutions of certain linear equations in $\beta\mathbb{Z}$. In particular, we show that the ability to guarantee the existence of $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ in one cell of a partition is equivalent to the ability to find idempotents $p$ and $q$ in $\beta\mathbb{N}$ such that $a_1\cdot p+a_2\cdot p+\cdots+a_n\cdot p=b_1\cdot q+b_2\cdot q+\cdots+b_m\cdot q$, and thus determine exactly when the latter has a solution.

AMS 2000 Mathematics subject classification: Primary 05D10. Secondary 22A15; 54H13