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

Let $r$, $m$ and $k\ge 2$ be positive integers such that $r\mid k$ and let $v\in [0,\lfloor \frac{k-1}{2r}\rfloor ]$ be any integer. For any integer $\mathrm{\ell }\in [1,k]$ and $ϵ\in \{0,1\}$, we let ${\mathcal{E}}_v^{(\mathrm{\ell },ϵ)}$ be the linear homogeneous equation defined by $$\mathcal{E}${v}^{(\ell, \epsilon)}: x_1 + \cdots + x{k-(rv+\epsilon)} =x_{k-(rv+\epsilon-1)} +\cdots+ \ell x_{k}$\mathrm{.}Wedenotethenumber$S_{\mathfrak{z},m}^{(\ell, \epsilon)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\chi: [1, t] \to {0, 1,\ldots,m-1}$,thereexistsasolution$(\hat{x}1, \hat{x}2, \ldots, \hat{x}k)$totheequation$$\mathcal{E}${v}^{(\ell,\epsilon)}$ that satisfies the $r$-zero-sum condition, namely, $\displaystyle\sum{i=1}^k\chi(\hat{x}i) \equiv 0\pmod{r}$\mathrm{.}Inthisarticle,wecompletelydeterminetheconstant$S{\mathfrak{z}, 2}^{(k,1)}(k;r;0)$,$S{\mathfrak{z}, m}^{(k-1,1)}(k;r;0)$,$S_{\mathfrak{z}, 2}^{(1,1)}(k;2;1)$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 $S_{\mathfrak{z},2}^{(1,1)}(k;2;v)$.