A Note on the Two-Color Rado Numbers for \(a(x – y) = bz\)

Abstract

Let \(R(a(x-y) = bz)\) denote the least integer \(n\) such that for every \(2\)-coloring of the set \(\{1, 2, \ldots, n\}\) there exists a monochromatic solution to \(a(x-y) = bz\). Recently, Gasarch, Moriarty, and Tumma conjectured that \(R(a(x-y) = bz) = b^2 + b + 1\), where \(1 < a < b\). In this note, we confirm this conjecture.