The Determination of 2-color zero-sum generalized Schur Numbers

TL;DR

This paper determines the exact value of a 2-color zero-sum generalized Schur number for a specific linear equation, providing a general formula for all relevant parameters.

Contribution

The paper solves a previously posed problem by deriving a general formula for the zero-sum Schur number when r divides k and k > r.

Findings

Exact value of S_{z,2}(k;r) = kr - 2r + 1 for all positive integers k, r with r | k and k > r.

Generalization of the zero-sum Schur number formula beyond the previously studied case r=4.

Provides a complete solution to the posed problem in the literature.

Abstract

Consider the equation $\mathcal{E}: x_1+ \cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\mid k$. 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 $\displaystyle \sum_{i=1}^k\chi(\hat{x}_i) \equiv 0\pmod{r}$. 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 $r \mid k$.