2018/06/29 by Leader, Imre, Russell, Paul A.
#05D10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1806.11374
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 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.