On Some Exact Formulas for $2$-Color Off-diagonal Rado Numbers

Source abstract

Let $\varepsilon_{0}$, $\varepsilon_{1}$ be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the $2$-color off-diagonal Rado number $R_2(\varepsilon_{0},\varepsilon_{1})$ to be the smallest $N$ such that for any 2-coloring of $[1,N]$, it must admit a monochromatic solution to $\varepsilon_{0}$ of the first color or a monochromatic solution to $\varepsilon_{1}$ of the second color. Mayers and Robertson gave the exact $2$-color off-diagonal Rado numbers $R_2(x+qy=z,x+sy=z).$ Xia and Yao established the formulas for $R_2(3x+3y=z,3x+qy=z)$ and $R_2(2x+3y=z,2x+2qy=z)$. In this paper, we determine the exact numbers $R_2(2x+qy=2z,2x+sy=2z)$, where $q, s$ are odd integers with $q>s\geq1$.