Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 130 / 3 / Solution

A rational matrix is a partition regular matrix when every finite coloring of the positive integers admits a monochromatic positive vector in its kernel. Its columns have the columns property if their indices can be partitioned into ordered nonempty blocks $B_1,\dots ,B_s$ such that the columns in $B_1$ sum to zero and, for $j>1$, the sum over $B_j$ lies in the rational linear span of the columns in the earlier blocks. Rado's theorem states that a rational matrix is partition regular if and only if its columns have this property.

For one equation, clear denominators and writeThe one-row columns property is equivalent to the existence of a nonempty $I\subseteq [n]$ with

First suppose such an $I$ exists. Choose $i_0\in I$, put $c=|c_{i_0}|$, $C={\sum }_{j\notin I}{c}_j$, and choose $p\ge |C|$. The monochromatic m-p-c set theorem, whose finite induction proof uses the Van der Waerden theorem, gives positive $z_1,z_2$ for which all numbersare positive and have one color. SetThe middle coefficient is the integer $-C\mathrm{sgn}(c_{i_0})$, of absolute value at most $p$, so all the $x_i$ belong to the monochromatic set. Their $z_1$ contribution vanishes because the coefficients over $I$ sum to zero, and their $z_2$ contribution isThus the equation is partition regular.

Conversely, suppose no nonempty subset of the coefficients sums to zero. Choose a prime number $p$ that divides none of the finitely many nonzero subset sums. Color each positive integer by its last nonzero digit coloring in base $p$. If a monochromatic solution existed, let $v$ be the smallest P-adic valuation among its coordinates and let $I$ index the coordinates of valuation $v$. After division by $p^v$ and reduction modulo $p$, all $x_i$ with $i\in I$ have the same nonzero last digit $r$, while the other terms vanish. The equation would givecontrary to the choice of $p$. This proves the Rado theorem for one equation.