The 2-color Rado Number of $x_1+x_2+\cdots +x_{m-1}=a{x}_m,$ 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 $x_1+x_2+\cdots +x_{m-1}=a{x}_m$ is $\lceil \frac{m-1}{a}\lceil \frac{m-1}{a}\rceil \rceil$. Here we obtain the best possible improvement of the bound on $m.$ 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 $m\ge 2a+1$ but not when $m=2a,$ and that if $3\nmid a$ 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 $m=2a+1.$ We also determine the 2-color Rado number for all $a\ge 3$ and $m\ge \frac{a}{2}+1.$

arXiv comment18 pages

References in corpus (1)

• The 2-color Rado number of $x_1+x_2+...+x_{m-1}=a{x}_m$