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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.

Rado's theorem states that a rational matrix is partition regular if and only if it has the columns property. By hypothesis each $A_i$ is partition regular, so choose a columns partition for each one. Form the block-diagonal matrixTaking at stage $j$ the union of the $j$th blocks from the individual partitions, with empty blocks added after a partition ends, gives the columns property for $B$: each row block sees exactly the corresponding dependence for its $A_i$. Hence $B$ is partition regular by Rado's theorem. A monochromatic vectorin the kernel of $B$ satisfies $A_i{x}_i=0$ for every $i$, and all entries of all the $x_i$ have the same color.

Solved by gpt-5.6-sol high.