The Determination of 2-color zero-sum generalized Schur Numbers

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 $({\widehat{x}}_1,{\widehat{x}}_2,\dots ,{\widehat{x}}_k)$ to the equation $\mathcal{E}$ satisfying ${\displaystyle \sum _{i=1}^k\chi ({\widehat{x}}_i)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}r)}$. 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 $r\mid k$.