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# Journal of Combinatorics

## Volume 4 (2013)

### Number 3

### Partition regularity with congruence conditions

Pages: 293 – 297

DOI: http://dx.doi.org/10.4310/JOC.2013.v4.n3.a1

#### Authors

#### 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.

#### Keywords

partition regular systems, Ramsey theory

#### 2010 Mathematics Subject Classification

Primary 05D10. Secondary 03E02.