Infinite partition regular matrices: solutions in central sets

Abstract:

A finite or infinite matrix $A$ is image partition regular provided that whenever ${\mathbb N}$ is finitely colored, there must be some $\vec {x}$ with entries from ${\mathbb N}$ such that all entries of $A\vec {x}$ are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that $A$ must have images in every central subset of $\mathbb {N}$. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever ${\mathbb N}$ is finitely colored, there must exist injective sequences $\langle x_n\rangle _{n=0}^\infty$ and $\langle z_n\rangle _{n=0}^\infty$ in ${\mathbb N}$ with all sums of the forms $x_n+x_m$ and $z_n+2z_m$ with $n<m$ in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.