For a positive integer $m$ and a real number $c$, let $R = R(m,c,2)$ denote the discrete 2-color Rado number for the equation $x_1 + x_2 + \dots + x_m + c = 2x_0$. In other words, $R$ is the smallest integer such that for any coloring of the integers ${1, 2, \dots, R}$, there exist numbers $x_1, x_2, \dots, x_m, x_0$, all with the same color, such that $x_1 + x_2 + \dots + x_m + c = 2x_0$. In this article we show that if $m \geq 2$ and $c > 0$, then $$ \begin{array}{cc} R(m,c,2) =
\begin{cases}
\infty & \text{for $m$ even, $c$ odd} \newline
\big\lceil \frac{m}{2} \big\lceil \frac{m+c}{2} \big\rceil + \frac{c}{2} \big\rceil & \text{otherwise.} \end{cases} \end{array} $$ For real numbers $a$ and $c$, we look at the 2-color Rado number for the equation $x_1 + c = ax_0$. We show that if $a > 1$ and $c > 0$, then the 2-color continuous Rado number is $$
R_\mathbb{R}(1, c, a) =
\begin{cases}
\infty & \text{if $a=1$ } \newline
\frac{c}{a-1} & \text{otherwise.}
\end{cases}
$$
From this, we will show that the discrete Rado number is
$$
R(1, c, a) =
\begin{cases}
\frac{c}{a-1} & \text{if $ \left( a-1 \right) \mid c$} \newline
\infty & \text{otherwise.}
\end{cases}
$$
Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
College of Liberal Arts and Sciences, Kitasato University（北里大学文理学院）；Department of Mathematical Sciences, Faculty of Science, Shinshu University（信州大学理学部数学科学系）
