= Scaling invariance of homogeneous partition regularity
{title2=$A(c\mathbf x)=cA\mathbf x$}
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.
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