The continuous two-color Rado numbers for \(x_1+x_2+\dots+x_m={ax}_0\). (English) Zbl 1314.05209
The paper is less than surprising. First (Theorem 1), for some evident 2-coloring of real numbers \(\geq 1\), the equation \(x_1 = ax_0, a \neq 1\) has no monochrome solutions. Second (Theorem 2), if \(a\), \(m\) are integers and \(a\geq 2\), \(m\geq a(a-1)\), the equation \(x_1 + \ldots + x_m = ax_0\) has a monochrome solution under each 2-coloring \([1, R]\) if and only if \(R\geq\frac{m^2}{a^2}\).
Reviewer: Igor Vladimirov Protasov (Kyiv)
MSC:
05D10 | Ramsey theory |