P-adic columns lemma

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.

After clearing denominators in a matrix, for every disjoint pair of index sets $I,B$ with $B$ nonempty and ${\sum }_{i\in B}{a}_i$ outside the linear span of the $I$-columns, choose an integer linear functional vanishing on those columns but not on the indicated sum. Choose a prime number dividing none of the finitely many nonzero evaluations. The last nonzero digit coloring for this prime, applied to a monochromatic positive solution, groups coordinates into blocks by P-adic valuation. Applying the appropriate functional and reducing modulo the prime shows that each block sum lies in the linear span of all earlier columns. This proves necessity in Rado's theorem without an asymptotic limiting argument.

For a prime number $p$, write a positive integer as $x=p^{v_p(x)}u$ with $p\nmid u$. Its last nonzero base-$p$ digit is $u\mathrm{mod}p$, an element of $\{1,\dots ,p-1\}$. Assigning this digit as the color gives the last nonzero digit coloring used in the necessity proof of Rado's theorem.