= Partition-regular system forcing an ordered Schur relation
{title2=$3x+y=3z,\quad x-2y=w>0$}
The positive-variable system $3x+y=3z$, $x-2y=w$ is <partition regular>. Its columns for $x,z,w$ sum to zero, and the remaining $y$ column lies in their rational span, satisfying the <columns condition>. Thus every finite colouring has monochromatic positive $x,y,z,w$ with $2y<x$. A triangular <m-p-c set> gives the explicit solution $x=3u$, $y=3v$, $z=3u+v$, $w=3u-6v$, with positivity guaranteed by membership in that set.
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