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

TL;DR

This paper establishes a connection between Rado numbers and Erdős-Ginzburg-Ziv type results for linear inequalities in three variables, showing that for many such inequalities, the minimal number ensuring certain solutions is equal to the classical 2-color Rado number.

Contribution

It proves that for linear inequalities in three variables, the minimal integer guaranteeing solutions with sum zero mod 3 equals the classical 2-color Rado number, extending Erdős-Ginzburg-Ziv type results.

Findings

R( ext{L}, ext{Z}/3 ext{Z})=R( ext{L}, 2) for many inequalities

Identification of families where such equalities do not hold

Extension of Erdős-Ginzburg-Ziv results to 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.