Color Rado Numbers for $\sum _{i=1}^{m-1}{x}_i+c=x_m$

Abstract

Rado numbers are closely related to Ramsey numbers, but pertaining to equations and integers instead of cliques within graphs. For every integer $m\ge 3$ and every integer $c$, let the 2-color Rado number $r(m,c)$ be the least integer, if it exists, such that for every 2-coloring of the set $\{1,2,\dots ,r(m,c)\}$ there exists a monochromatic solution to the equation $\sum _{i=1}^{m-1}{x}_i+c=x_m$ .The values of $r(m,c)$ have been determined previously for nonnegative values of $c$, as well as all values of $m$ and $c$ such that $-m+2<c<0$ and $c<-(m-1)(m-2)$. In this paper, we find $r(m,c)$ for the remaining values of $m$ and $c$.