Two Color Off-diagonal Rado-type Numbers

Abstract

We show that for any two linear homogenous equations ${\mathcal{E}}_0,{\mathcal{E}}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of ${\mathbb{Z}}^{+}$ admits monochromatic solutions of color 0 to ${\mathcal{E}}_0$ or monochromatic solutions of color 1 to ${\mathcal{E}}_1$. We define the 2-color off-diagonal Rado number $RR({\mathcal{E}}_0,{\mathcal{E}}_1)$ to be the smallest $N$ such that $[1,N]$ must admit such solutions. We determine a lower bound for $RR({\mathcal{E}}_0,{\mathcal{E}}_1)$ in certain cases when each ${\mathcal{E}}_i$ is of the form $a_1{x}_1+...+a_n{x}_n=z$ as well as find the exact value of $RR({\mathcal{E}}_0,{\mathcal{E}}_1)$ when each is of the form $x_1+a_2{x}_2+...+a_n{x}_n=z$. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.

10 pages