EGZ-generalizations for linear equations and linear inequalities in three variables

For a Diophantine system of equalities or inequalities in $k$ variables, $\mathcal{L}$, we denote by $R(\mathcal{L}, r)$ the classical \emph{$r$-color Rado number}, that is, $R(\mathcal{L}, r)$ is the smallest integer, if it exist, such that for every $r$-coloring of $[1,R(\mathcal{L}, r)]$ there exist a monochromatic solution of $\mathcal{L}$. In 2003 Bialostocki, Bialostocki and Schaal studied the related parameter, $R(\mathcal{L}, \Z_r)$, defined as the smallest integer, if it exist, such that for every $(\Z_r)$-coloring of $[1,R(\mathcal{L}, \Z_r)]$ there exist a zero-sum solution of $\mathcal{L}$; in view of the Erd\H{o}s-Ginzburg-Ziv theorem, the authors state that the system $\mathcal{L}$ admits an EGZ-generalization if $R(\mathcal{L}, 2)= R(\mathcal{L}, \Z/k\Z)$. In this work we we prove that any linear inequality on three variables, $$\mathcal{L}_3: ax+by+cz+d<0,$$ where $a,b,c,d\in\Z$ with $abc\neq 0$, admits an EGZ-generalization except in the cases where there is no positive solution of the inequality. More over, we determine the corresponding $2$-color Rado numbers depending on the coefficients of $\mathcal{L}_3$. This is joint work with Amanda Montejano.Non UBCUnreviewedAuthor affiliation: CONACyT/UAZFacult