Scaling invariance of homogeneous partition regularity ($A(c\mathbf{x})=cA\mathbf{x}$)

For any positive integer $c$, a homogeneous rational matrix is partition regular over $c\mathbb{N}$ if and only if it is partition regular over $\mathbb{N}$. Pull a colouring of $c\mathbb{N}$ back along $n\mapsto cn$ and scale the resulting solution. Conversely, restrict a colouring of all positive integers to $c\mathbb{N}$. The argument depends on homogeneity and does not automatically extend to inhomogeneous equations or prescribed distinctness conditions.