Partition regularity with congruence conditions

TL;DR

This paper investigates whether additional divisibility constraints can be imposed on solutions to image partition regular matrices and shows that such constraints are generally not possible, also disproving a related conjecture.

Contribution

It demonstrates that imposing divisibility conditions on solutions to image partition regular matrices is not always feasible and refutes a conjectured equivalence between two main forms of partition regularity.

Findings

Divisibility constraints cannot always be imposed on solutions.

Disproves the conjectured equivalence between image and kernel partition regularity.

Abstract

An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist that each variable x_i is a multiple of some given d_i? This is a question of Hindman, Leader and Strauss. Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity.