= Partition regularity dichotomy under an extra constraint
For a <partition regular matrix> $A$ and any property $P$ of its positive solution vectors, either every <finite coloring> admits a <monochromatic> solution with $P$, or every <finite coloring> admits one without $P$. If the first alternative fails, take a witnessing coloring and pair it with any other coloring. Partition regularity of the product coloring forces a solution avoiding $P$ in the other coloring. The alternatives need not be exclusive.
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