= 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 $B_1,\ldots,B_s$ 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 write
$$
c_1x_1+\cdots+c_nx_n=0,
\qquad c_i\in\mathbb Z\setminus\{0\}.
$$
The one-row columns property is equivalent to the existence of a nonempty $I\subseteq[n]$ with
$$
\sum_{i\in I}c_i=0.
$$
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 $p\geq|C|$. 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 numbers
$$
cz_1+\lambda z_2\quad(|\lambda|\leq p),
\qquad cz_2
$$
are positive and have one color. Set
$$
x_i=cz_1\quad(i\in I\setminus\{i_0\}),
\qquad
x_{i_0}=cz_1-\frac{Cc}{c_{i_0}}z_2,
\qquad
x_j=cz_2\quad(j\notin I).
$$
The middle coefficient is the integer $-C\operatorname{sgn}(c_{i_0})$, 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 is
$$
c_{i_0}\left(-\frac{Cc}{c_{i_0}}\right)+Cc=0.
$$
Thus 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 give
$$
r\sum_{i\in I}c_i\equiv0\pmod p,
$$
contrary to the choice of $p$. This proves the <Rado theorem for one equation>.
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