The $2$-Color Rado Number of $x_1+x_2+\dots +x_{m-1}=a{x}_m$

Abstract

In 1982, Beutelspacher and Brestovansky proved that for every integer $m\ge 3$, the $2$-color Rado number of the equation
$$x_1+x_2+\dots +x_{m-1}=x_m$$
is $m^2–m–1$. In 2008, Schaal and Vestal proved that, for every $m\ge 6$, the $2$-color Rado number of
$$x_1+x_2+\dots +x_{m-1}=2x_m$$
is $\lceil \frac{m-1}{2}\lceil \frac{m-1}{2}\rceil \rceil$. Here, we prove that, for every integer $a\ge 3$ and every $m\ge 2a^2–a+2$, the 2-color Rado number of
$$x_1+x_2+\dots +x_{m-1}=a{x}_m$$
is $\lceil \frac{m-1}{a}\lceil \frac{m-1}{a}\rceil \rceil$. For the case $a=3$, we show that our formula gives the Rado number for all $m\ge 7$, and we determine the Rado number for all $m\ge 3$.