Progressions in every two-coloration of \(Z_ n\) (Q1199996)

Defie \(G(n)\) to be the largest integer such that in any two colouring of \(\mathbb{Z}/n\mathbb{Z}\), there is a \(G(n)\) term monochromatic arithmetic progression of distinct elements. The main theorem here implies that \(G(n)\leq G(rn)\leq\max\{r,G(n)\}\) unless, in any two colouring of \(\mathbb{Z}/n\mathbb{Z}\) where the longest monochromatic arithmetic progression of distinct elements has exactly \(G(n)\) elements, we have that the \(G(n)+1\)st element always coincides with the first element (which seems very unlikely). The authors also point out and use the connection between \(G(n)\) and the van der Waerden numbers \(W(k)\), so that \(G(n)\geq k\) if \(n\geq W(k)\).