 Columns partition for finite-sums systems

Introduce a variable $z_I$ for every nonempty $I\subseteq [k]$ and the homogeneous equations $z_I-{\sum }_{i\in I}{z}_{\{i\}}=0$ for $|I|\ge 2$. In their matrix $A$, partition the columns by the displayed blocks. For each $j$, the vector with coordinate $1$ exactly when $j\in I$ lies in the kernel of a linear map of $A$. Its nonzero coordinates occur in $B_j$ and earlier blocks only; every coordinate in $B_j$ is one. Hence the first block's column sum is zero and every subsequent block sum is in the linear span of earlier columns. The columns condition holds, so Rado's theorem forces a positive monochromatic solution, and all nonempty finite sums of its singleton coordinates have the same colour.