= Compactness bound for partition regularity
If a <rational matrix> is a <partition regular matrix>, then for each fixed number of colors $k$ some finite <integer interval> $[T]$ already forces a <monochromatic> positive solution. Otherwise the solution-free <finite colorings> of successive intervals form a finitely branching <tree> with every level nonempty. The <König infinity lemma> gives an infinite branch, contradicting partition regularity. More generally, this argument applies to any family of configurations each using finitely many <positive integers>.
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