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 integer 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. In this paper, we establish two exact formulas of R 2(3x + 3y = z, 3x + 3qy = z) and R 2(2x + 3y = z, 2x + 2qy = z).
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Yao, O.X.M., Xia, E.X.W. Two Formulas of 2-Color Off-Diagonal Rado Numbers. Graphs and Combinatorics 31, 299–307 (2015). https://doi.org/10.1007/s00373-013-1378-9
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DOI: https://doi.org/10.1007/s00373-013-1378-9