Journal of Combinatorics
Volume 4 (2013)
Partition regularity with congruence conditions
Pages: 293 – 297
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.
partition regular systems, Ramsey theory
2010 Mathematics Subject Classification
Primary 05D10. Secondary 03E02.