Partition regular matrix

A rational matrix $A$ is partition regular when every finite coloring of the positive integers admits a nonzero monochromatic vector $x$ satisfying $Ax=0$.

If $c_1,\dots ,c_n$ are the columns of a matrix, the matrix has the columns property when $[n]$ has an ordered partition $B_1\bigsqcup \cdots \bigsqcup B_s$ such thatand, for $j>1$, ${\sum }_{i\in B_j}{c}_i$ belongs to the linear span of the columns indexed by $B_1\cup \cdots \cup B_{j-1}$.

Rado's theorem states that a rational matrix is partition regular if and only if it has the columns property.

The necessity direction in Rado's theorem colors an integer using initial data from a P-adic valuation. Applying partition regularity and grouping the coordinates of a monochromatic solution by valuation yields the blocks in the columns property; reduction modulo successively higher powers of $p$ supplies the required linear dependences.