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 \(\Delta: \{1, 2, \ldots, 2n\} \to \{0, 1\}\),
there exists a solution \((x_1, x_2, x_3)\) to
\[x_1 + x_2 + x_3 = c\]
such that \(x_i \neq x_j\) when \(i \neq j\),
and
\(\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\).

In this paper, it is shown that for every integer \(c\),
\[R_d(c) =
\begin{cases}
4c + 8 & \text{if } c \geq 1,\\
8 & \text{if } -3 \leq c < -6,\\ 9 & \text{if} c=0,-2,-7,-8\\ 10 & \text{if } c =-1,-9 \\ |c| -\left\lfloor \frac{|c|-4}{5} \right\rceil & \text{if } c \leq -10. \end{cases}\]