Determination of the two-color Rado number for $a_1x_1+...+a_mx_m=x_0$

Abstract

For positive integers $a_1,a_2,...,a_m$, we determine the least positive integer $R(a_1,...,a_m)$ such that for every 2-coloring of the set $[1,n]={1,...,n}$ with $n\ge R(a_1,...,a_m)$ there exists a monochromatic solution to the equation $a_1x_1+...+a_mx_m=x_0$ with $x_0,...,x_m\in[1,n]$. The precise value of $R(a_1,...,a_m)$ is shown to be $av^2+v-a$, where $a=min{a_1,...,a_m}$ and $v=\sum_{i=1}^{m}a_i$. This confirms a conjecture of B. Hopkins and D. Schaal.