 

Neil Hindman and Kendra Pleasant
Preserving positive integer images of matrices view print


Published: 
September 17, 2016

Keywords: 
Image partition regular, weakly image partition regular, Ramsey Theory, images of integer matrices 
Subject: 
Primary: 05D10; Secondary 15A24 


Abstract
We prove that whenever u,v∈N and
A is a u× v matrix with integer entries and rank n,
there is a u× n matrix B such that
{A\vec k:\vec k∈Z^{v}}∩ N^{u}={B\vec x:\vec x∈N^{n}}∩N^{u}.
As a consequence we obtain the following result which answers a question
of Hindman, Leader, and Strauss: Let R be a subring of the
rationals with 1∈ R and let S={x∈ R:x>0}. If A is a finite matrix with rational entries,
then there is a matrix B with no more columns than A such that the set of images of
B in S via vectors with entries from S is exactly the same as as the set of
images of A in S via vectors with entries from R.
We also show that the notion of image partition regularity is strictly
stronger than that of weak image partition regularity in terms of
Ramsey Theoretic consequences. That is, we show that for each
u≧ 3, there are no v and a
u× v matrix A such that for any \vec y∈
{A\vec k:\vec k∈Z^{v}}∩N^{u}, the set of entries
of \vec y form (in some order) a length u arithmetic progression.


Acknowledgements
The first author acknowledges support received from the National Science Foundation (USA) via Grant DMS1460023.


Author information
Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA.
nhindman@aol.com
Kendra Pleasant:
Department of Mathematics. Howard University, Washington, DC 20059, USA.
kpleasant90@gmail.com

