= Columns partition for finite-sums systems
{title2=$B_j=\{I:\min I=j\}$}
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 $|I|\ge2$. 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.
Back to article page
