  Consider the equation $\mathcal{E}: x_1+ \cdots+x_{k-1} =x_{k}$ and let $k$
and $r$ be positive integers such that $r\mid k$. The number
$S_{\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such
that for any 2-coloring $\chi: [1, t] \to \{0, 1\}$ there exists a solution
$(\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)$ to the equation $\mathcal{E}$
satisfying $\displaystyle \sum_{i=1}^k\chi(\hat{x}_i) \equiv 0\pmod{r}$. In a
recent paper, the first author posed the question of determining the exact
value of $S_{\mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and
show, more generally, that $S_{\mathfrak{z}, 2}(k, r)=kr - 2r+1$ for all
positive integers $k$ and $r$ with $k>r$ and $r \mid k$.
