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

TL;DR

This paper generalizes the 2-color Rado number for the equation involving a parameter a, providing a formula valid for all sufficiently large m and specific cases, extending previous results for particular values of a.

Contribution

The paper derives a general formula for the 2-color Rado number of the equation with parameter a, valid for all m beyond a certain threshold, extending prior specific cases.

Findings

The Rado number formula is eil((m-1)/a) \u001ceil((m-1)/a)eil((m-1)/a)eil((m-1)/a) for all m t least 2a^2 - a + 2.

For a=3, the formula applies for all m  7, and the Rado number is explicitly determined for all m  3.

The results unify and extend previous findings for specific equations with a=1 and a=2.

Abstract

In 1982, Beutelspacher and Brestovansky proved that for every integer $m\geq 3,$ 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 $m\geq 6,$ 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 $a\geq 3$ and every $m\geq 2a^2-a+2$, the 2-color Rado number of $$x_1+x_2+...+x_{m-1}=ax_m$$ 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 $m\geq 7,$ and we determine the Rado number for all $m\geq 3.$