On Some Exact Formulas for \(2\)-Color Off-diagonal Rado Numbers

Let \(ε₀\), \(ε₁\) 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 \(R_2(ε₀,ε₁)\) to be the smallest \(N\) such that for any 2-coloring of \([1,N]\), it must admit a monochromatic solution to \(ε₀\) of the first color or a monochromatic solution to \(ε₁\) 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 \(q>s≥1\).