The \(2\)-color Rado Number of \(x + y + kz = 3w\)

Abstract

For any positive integer \(k\), there exists a smallest positive integer \(N\), depending on \(k\), such that every \(2\)-coloring of \(1, 2, \ldots, N\) contains a monochromatic solution of the equation \(x + y + kz = 3w\). Based on computer checks, Robertson and Myers in \([5]\) conjectured values for \(N\) depending on the congruence class of \(k\) (mod \(9\)). In this note, we establish the values of \(N\) and find that in some cases they depend on the congruence class of \(k\) (mod \(27\)).