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

Let \ (₀\), \ (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 \ (R₂ (₀, ₁) \) to be the smallest \ (N\) such that for any 2-coloring of \ (1, N\), it must admit a monochromatic solution to \ (₀\) of the first color or a monochromatic solution to \ (₁\) of the second color. Mayers and Robertson gave the exact \ (2\) -color off-diagonal Rado numbers \ (R₂ (x+qy=z, x+sy=z). \) Xia and Yao established the formulas for \ (R₂ (3x+3y=z, 3x+qy=z) \) and \ (R₂ (2x+3y=z, 2x+2qy=z) \). In this paper, we determine the exact numbers \ (R₂ (2x+qy=2z, 2x+sy=2z) \), where \ (q, s\) are odd integers with \ (q>s1\).