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

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          Abstract

          Let \(r\), \(m\) and \(k\geq 2\) 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 \(\chi: [1, t] \to \{0, 1,\ldots,m-1\}\), there exists a solution \((\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)\) to the equation \(\mathcal{E}_{v}^{(\ell,\epsilon)}\) that satisfies the \(r\)-zero-sum condition, namely, \(\displaystyle\sum_{i=1}^k\chi(\hat{x}_i) \equiv 0\pmod{r}\). 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 \(S_{\mathfrak{z},2}^{(1,1)}(k;2;v)\).

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          Most cited references 3

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          Zero-sum problems in finite abelian groups: A survey

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            Zero-sum problems — A survey

             Yair Caro (1996)
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              On the Erdős-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

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                Author and article information

                Journal
                27 August 2018
                Article
                1808.08725

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.CO

                Combinatorics

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