Maximality of Infinite Partition Regular Matrices

TL;DR

This paper investigates the maximality and extension properties of infinite image partition regular matrices, revealing new compatibility results for Milliken-Taylor systems using algebraic tools from Stone-Cech compactification.

Contribution

It provides new results on when infinite matrices can be extended while remaining image partition regular, including surprising compatibility results for Milliken-Taylor systems.

Findings

Milliken-Taylor systems can be adjoined while maintaining image partition regularity.

Extensions of infinite systems are possible under certain conditions.

Conjectures are proposed regarding maximality and extensions of these matrices.

Abstract

A finite or infinite matrix $A$ with rational entries (and only finitely many non-zero entries in each row) is called image partition regular if, whenever the natural numbers are finitely coloured, there is a vector $x$, with entries in the natural numbers, such that $Ax$ is monochromatic. Many of the classical results of Ramsey theory are naturally stated in terms of image partition regularity. Our aim in this paper is to investigate maximality questions for image partition regular matrices. When is it possible to add rows on to $A$ and remain image partition regular? When can one add rows but `nothing new is produced'? What about adding rows and also new variables? We prove some results about extensions of the most interesting infinite systems, and make several conjectures. Perhaps our most surprising positive result is a compatibility result for Milliken-Taylor systems, stating that (in many cases) one may adjoin one Milliken-Taylor system to a translate of another and remain image partition regular. This is in contrast to earlier results, which had suggested a strong inconsistency between different Milliken-Taylor systems. Our main tools for this are some algebraic properties of the $\beta N$, the Stone-Cech compactification of the natural numbers.