On Generalized Schur Numbers for \(x_1+x_2+c=kx_3\)

Abstract

For every integer \(c\) and every positive integer \(k\), let \(n = r(c, k)\) be the least integer, provided that it exists, such that for every coloring

\[\Delta: \{1,2,\ldots,n\} \rightarrow \{0,1\},\]

there exist three integers, \(x_1, x_2, x_3\), (not necessarily distinct) such that
\[\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\]
and
\[x_1+x_2+c= kx_3.\]

If such an integer does not exist, then let \(r(c, k) = \infty\). The main result of this paper is that

\[r(c,2) =
\begin{cases}
|c|+1 & \text{if } c \text{ is even} \\
\infty & \text{if } c \text{ is odd}
\end{cases}\]

for every integer \(c\). In addition, a lower bound is found for \(r(c, k)\) for all integers \(c\) and positive integers \(k\) and linear upper and lower bounds are found for \(r(c, 3)\) for all positive integers \(c\).