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Continuous Rado numbers for the equation \(a_1x_1+a_2x_2+\cdots+a_{m-1}x_{m-1}+c=x_m\). (English) Zbl 1258.05123

Let \(a_1,a_2,\dots,a_{m-1}\) (\(m \geq 3 \)) be positive integers, and let \(a = \min\{a_1,\dots,a_{m-1}\}\) and \(v = \sum_{k=1}^{m-1} a_k\). S. Guo and the reviewer [J. Comb. Theory, Ser. A 115, No.2, 345–353 (2008; Zbl 1133.05097)] confirmed a conjecture of B. Hopkins and D. Schaal [Adv. in Appl. Math. 152, No. 4, 433–441 (2001; Zbl 1091.05069)] by proving that the Rado number \(r(a_1,\dots,a_{m-1})\), which is the least positive integer \(n\) such that for any \(2\)-coloring of \(1,2,\dots,n\) the equation
\[ a_1x_1+\dots+a_{m-1}x_{m-1} = x_m \]
has a monochromatic solution \(x_1,\dots,x_m \in \{1,\dots,n\}\), coincides with \(av^2+v-a\).
In the paper under review, for any real number \(c\) the authors define the \(2\)-color continuous Rado number \(r_{\mathbb{R}}(a_1,\dots,a_{m-1},c)\) as the least real number \(r\) such that for each \(2\)-coloring of the interval \([1,r] = \{x\in\mathbb{R} : 1 \leq x \leq r\}\) the equation
\[ a_1x_1+\dots+a_{m-1}x_{m-1}+c=x_m \]
has a monochromatic solution \(x_1,\dots,x_m \in [1,r]\). The authors show that \[ r_{\mathbb{R}}(a_1,\dots,a_{m-1},c) = av^2+v-a+c(av+a+1) \]
if \(c \geq 1-v\), and that
\[ r_{\mathbb{R}}(a_1,\dots,a_{m-1},c) = \frac{1-c(av+a+1)}{av^2+v-a} \]
if \(c<1-v\). When \(a_1 = \dots = a_{m-1}=1\) and \(c \geq1-v = 2-m\), this is an earlier result of C. S. Brady and R. Haas [Congr. Numerantium 177, 109–114 (2005; Zbl 1089.05076)].

MSC:

05D10 Ramsey theory
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