On some exact formulas for \(2\)-color off-diagonal Rado numbers. (English) Zbl 07888105
Summary: 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. K. Myers and A. Robertson [Electron. J. Comb. 14, No. 1, Research Paper R53, 10 p. (2007; Zbl 1157.05336)] gave the exact \(2\)-color off-diagonal Rado numbers \(R_2(x+qy=z,x+sy=z)\). O. X. M. Yao and E. X. W. Xia [Graphs Comb. 31, No. 1, 299–307 (2015; Zbl 1308.05101)] 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 \(q>s\geq1\).
MSC:
| 05D10 | Ramsey theory |
References:
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