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

Abstract

In the first installment of this series, we proved 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 $\lceil \frac{m-1}{a}\lceil \frac{m-1}{a}\rceil \rceil$. Here, we obtain the best possible improvement of the bound on $m$. Specifically, we prove that if $3|a$, then the $2$-color Rado number is $\lceil \frac{m-1}{a}\lceil \frac{m-1}{a}\rceil \rceil$ when $m\ge 2a+2$ but not when $m=2a+1$, and that if $3\nmid$ is composite, then the $2$-color Rado number is $\lceil \frac{m-1}{a}\lceil \frac{m-1}{a}\rceil \rceil$ when $m\ge 2a+2$ but not when $m=2a+1$. Additionally, we determine the $2$-color Rado number for all $a\ge 3$ and $m\ge \frac{a}{3}+1$.