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Register a new userThe Determination of 2-color zero-sum generalized Schur Numbers
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Consider the equation $mathcal{E}: x_1+ cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $rmid k$. The number $S_{mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $chi: [1, t] to {0, 1}$ there exists a solution $(hat{x}_1, hat{x}_2, ldots, hat{x}_k)$ to the equation $mathcal{E}$ satisfying $displaystyle sum_{i=1}^kchi(hat{x}_i) equiv 0pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and show, more generally, that $S_{mathfrak{z}, 2}(k, r)=kr - 2r+1$ for all positive integers $k$ and $r$ with $k>r$ and $r mid k$.
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The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Omega subset B(G$), let $d_{Omega}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|geq t$ has a subsequence in $Omega$.We provide some first results and open problems on $d_{Omega}(G)$.
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