On determination of Zero-sum $\ell$-generalized Schur Numbers for some linear equations

Abstract

Let $r$, $m$ and $k\ge 2$ be positive integers such that $r\mid k$ and let $v\in \left[0,\lfloor \frac{k-1}{2r}\rfloor \right]$ be any integer. For any integer $\ell \in [1,k]$ and $\epsilon \in \{0,1\}$, we let ${\mathcal{E}}_v^{(\ell ,\epsilon )}$ be the linear homogeneous equation defined by ${\mathcal{E}}_v^{(\ell ,\epsilon )}:x_1+\cdots +x_{k-(rv+\epsilon )}=x_{k-(rv+\epsilon -1)}+\cdots +\ell x_k$. We denote the number $S_{\mathfrak{z},m}^{(\ell ,\epsilon )}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\chi :[1,t]\to \{0,1,\dots ,m-1\}$, there exists a solution $({\hat{x}}_1,{\hat{x}}_2,\dots ,{\hat{x}}_k)$ to the equation ${\mathcal{E}}_v^{(\ell ,\epsilon )}$ that satisfies the $r$-zero-sum condition, namely, $\sum _{i=1}^k\chi ({\hat{x}}_i)\equiv 0(\mathrm{mod}r)$. In this article, we completely determine the constant $S_{\mathfrak{z},2}^{(k,1)}(k;r;0)$, $S_{\mathfrak{z},m}^{(k-1,1)}(k;r;0)$, $S_{\mathfrak{z},2}^{(1,1)}(k;2;1)$ and $S_{\mathfrak{z},r}^{(1,0)}(k;r;v)$. Also, we prove upper bound for the constants $S_{\mathfrak{z},2}^{(2,1)}(k;2;0)$ and $S_{\mathfrak{z},2}^{(1,1)}(k;2;v)$.