Some multivariable Rado numbers

Abstract

The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $R_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of $[1,R_r(\mathcal{E})]$ must admit a monochromatic solution to $\mathcal{E}$. In this paper, we give upper and lower bounds for the Rado number of ${\sum }_{i=1}^{m-2}{x}_i+k{x}_{m-1}=\ell x_m$, and some exact values are also given. Furthermore, we derive some results for the cases that $\ell =m=4$ and $m=5,\ell =k+i(1\le i\le 5)$. As a generalization, the \emph{$r$-color Rado numbers} for linear equations ${\mathcal{E}}_1,{\mathcal{E}}_2,...,{\mathcal{E}}_r$ is defined as the minimal integer, if it exists, such that any $r$-coloring of $[1,R_r({\mathcal{E}}_1,{\mathcal{E}}_2,...,{\mathcal{E}}_r)]$ must admit a monochromatic solution to some ${\mathcal{E}}_i$, where $1\le i\le r$. A lower bound for $R_r({\mathcal{E}}_1,{\mathcal{E}}_2,...,{\mathcal{E}}_r)$ and the exact values of $R_2(x+y=z,\ell x=y)=5k$ and $R_2(x+y=z,x+a=y)$ was given by Lovász Local Lemma.