Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers

Abstract

Let $S_{\mathfrak{z}}(k,r)$ be the least positive integer such that for any $r$-coloring $\chi :\{1,2,\dots ,S_{\mathfrak{z}}(k,r)\}⟶\{1,2,\dots ,r\}$, there is a sequence $x_1,x_2,\dots ,x_k$ such that ${\sum }_{i=1}^{k-1}{x}_i=x_k$, and ${\sum }_{i=1}^k\chi (x_i)\equiv 0(\mathrm{mod}r)$. We show that when $k$ is greater than $r$, $kr-r-1\le S_{\mathfrak{z}}(k,r)\le kr-1$, and when $r$ is an odd prime, $S_{\mathfrak{z}}(k,r)$ is in fact equal to $kr-r$.

19 pages