Zero-sum Generalized Schur Numbers

Abstract: Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\chi:[1,n] \rightarrow {0,1,\dots,r-1}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$ with $\sum_{i=1}^{k} \chi(x_i) \equiv 0 \,(\mathrm{mod }\,r)$. We give some formulas and lower bounds for various instances.