Journal of Combinatorics

Volume 8 (2017)

Number 4

Duality for image and kernel partition regularity of infinite matrices

Pages: 653 – 672

DOI: https://dx.doi.org/10.4310/JOC.2017.v8.n4.a5

Authors

Neil Hindman (Department of Mathematics, Howard University, Washington, D.C., U.S.A.)

Imre Leader (Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Dona Strauss (Department of Pure Mathematics, University of Leeds, United Kingdom)

Abstract

A matrix $A$ is image partition regular over $\mathbb{Q}$ provided that whenever $\mathbb{Q} \backslash \lbrace 0 \rbrace$ is finitely coloured, there is a vector $\vec{x}$ with entries in $\mathbb{Q} \backslash \lbrace 0 \rbrace$ such that the entries of $A\vec{x}$ are monochromatic. It is kernel partition regular over $\mathbb{Q}$ provided that whenever $\mathbb{Q} \backslash \lbrace 0 \rbrace$ is finitely coloured, the matrix has a monochromatic member of its kernel. We establish a duality for these notions valid for both finite and infinite matrices. We also investigate the extent to which this duality holds for matrices partition regular over proper subsemigroups of $\mathbb{Q}$.

Keywords

kernel partition regularity, image partition regularity, infinite matrices

2010 Mathematics Subject Classification

05D10

Received 10 September 2015

Published 17 July 2017