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

TL;DR

This paper improves bounds on the 2-color Rado number for the equation x_1 + ... + x_{m-1} = a x_m, providing exact values for a wide range of parameters and refining previous results.

Contribution

It refines the bounds on the 2-color Rado number for the equation, determining exact values for broader parameter ranges and improving previous bounds.

Findings

Exact 2-color Rado numbers for m ≥ 2a+1 when 3 divides a

Exact 2-color Rado numbers for m ≥ 2a+2 when 3 does not divide a

Determination of Rado numbers for all a ≥ 3 and m ≥ (a/2)+1

Abstract

In the first installment of this series, we proved that, for every integer $a\geq 3$ and every $m\geq 2a^2-a+2$, the 2-color Rado number of $x_1 + x_2 + \cdots + x_{m-1} = ax_m$ 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 $m\geq 2a+1$ 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 $m\geq 2a+2$ but not when $m=2a+1.$ We also determine the 2-color Rado number for all $a\geq 3$ and $m\geq \frac{a}{2}+1.$