Distinct Rado Numbers for $x_1+x_2+c=x_3$

Abstract

For every integer $c$, let $n=R_d(c)$ be the least integer such
that for every coloring $\mathrm{\Delta }:\{1,2,\dots ,2n\}\to \{0,1\}$,
there exists a solution $(x_1,x_2,x_3)$ to
$$x_1+x_2+x_3=c$$
such that $x_i\ne x_j$ when $i\ne j$,
and
$\mathrm{\Delta }(x_1)=\mathrm{\Delta }(x_2)=\mathrm{\Delta }(x_3)$.

In this paper, it is shown that for every integer $c$,
$$R_d(c)=\{\begin{array}{ll}4c+8& \text{if }c\ge 1,\\ 8& \text{if }-3\le c<-6,\\ 9& \text{if}c=0,-2,-7,-8\\ 10& \text{if }c=-1,-9\\ |c|-\lfloor \frac{|c|-4}{5}\rceil & \text{if }c\le -10.\end{array}$$