= One-equation partition regularity over odd integers
{title2=$\sum_i a_i=0$}
A rational homogeneous equation $\sum_i a_ix_i=0$ has a <monochromatic> positive odd solution in every <finite colouring> of the odd integers exactly when its full coefficient sum is zero. Sufficiency uses a constant odd vector. For necessity, clear denominators and, if the total $S$ is nonzero, colour by odd <residue classes> modulo $2^K>|S|$. A common odd residue is invertible, so a solution would require $2^K\mid S$, impossible. This is stronger than the zero-sum-subset criterion for unrestricted <partition regularity>.
Back to article page
