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 $a_1x_1+...+a_nx_n=z$ as well as find the exact value of $RR(\mathcal{E}_0,\mathcal{E}_1)$ when each is of the form $x_1+a_2x_2+...+a_nx_n=z$. 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.