We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved that the system $Ax = b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general (commutative) ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new 'direct' proof of Rado’s result.
@article{10_37236_7972,
author = {Imre Leader and Paul A. Russell},
title = {Inhomogeneous partition regularity},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {2},
doi = {10.37236/7972},
zbl = {1444.05141},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7972/}
}
TY - JOUR
AU - Imre Leader
AU - Paul A. Russell
TI - Inhomogeneous partition regularity
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/7972/
DO - 10.37236/7972
ID - 10_37236_7972
ER -
%0 Journal Article
%A Imre Leader
%A Paul A. Russell
%T Inhomogeneous partition regularity
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/7972/
%R 10.37236/7972
%F 10_37236_7972
Imre Leader; Paul A. Russell. Inhomogeneous partition regularity. The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/7972
