Zero-sum Generalized Schur Numbers

Abstract

Let $r$ and $k$ be positive integers with $r\mid k$. Denote by $S_{\mathfrak{z}}(k;r)$ the minimum integer $n$ such that every coloring $\chi :[1,n]\to \{0,1,\dots ,r-1\}$ admits a solution to ${\sum }_{i=1}^{k-1}{x}_i=x_k$ with ${\sum }_{i=1}^k\chi (x_i)\equiv 0(\mathrm{mod}r)$. We give some formulas and lower bounds for various instances.

Cited by in corpus (2)

• The Determination of 2-color zero-sum generalized Schur Numbers
• Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers