= Reciprocal partition regularity
If a <rational matrix> $A$ is a <partition regular matrix>, every <finite coloring> of the <positive integers> admits <monochromatic> $y_1,\ldots,y_n$ such that
$$
A(1/y_1,\ldots,1/y_n)^{\mathsf T}=0.
$$
Choose a <compactness bound for partition regularity> $T$ for the number of colors, put $S=\operatorname{lcm}(1,\ldots,T)$, and pull back the coloring by $t\mapsto S/t$ on $[T]$. A <monochromatic> positive solution $x$ of $Ax=0$ gives $y_i=S/x_i$; then $(1/y_i)=x_i/S$. The <least common multiple> guarantees that every $y_i$ is a <positive integer>.
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