Erd\H{o}s-Ginzburg-Ziv type generalizations for linear equations and linear inequalities in three variables

Abstract

For any linear inequality in three variables $\mathcal{L}$, we determine (if it exist) the smallest integer $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})$ such that: for every mapping $\chi :[1,n] \to \{0,1,2\}$, with $n\geq R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})$, there is a solution $(x_1,x_2,x_3)\in [1,n]^3$ of $\mathcal{L}$ with $\chi(x_1)+\chi(x_2)+\chi(x_3)\equiv 0$ (mod $3$). Moreover, we prove that $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})=R(\mathcal{L}, 2)$, where $R(\mathcal{L}, 2)$ denotes the classical $2$-color Rado number, that is, the smallest integer (provided it exist) such that for every $2$-coloring of $[1,n]$, with $n\geq R(\mathcal{L}, 2)$, there exist a monochromatic solution of $\mathcal{L}$. Thus, we get an Erd\H{o}s-Ginzburg-Ziv type generalization for all lineal inequalities in three variables having a solution in the positive integers. We also show a number of families of linear equations in three variables $\mathcal{L}$ such that they do not admit such Erd\H{o}s-Ginzburg-Ziv type generalization, named $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})\neq R(\mathcal{L}, 2)$. At the end of this paper some questions are proposed.