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

Abstract

In 1982, Beutelspacher and Brestovansky proved that for every integer $m\ge 3,$ the 2-color Rado number of the equation $$x_1+x_2+...+x_{m-1}=x_m$$ is $m^2-m-1.$ In 2008, Schaal and Vestal proved that, for every $m\ge 6,$ the 2-color Rado number of $$x_1+x_2+...+x_{m-1}=2x_m$$ 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 $$x_1+x_2+...+x_{m-1}=a{x}_m$$ 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 $m\ge 3.$

15 pages

Cited by in corpus (1)

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