On the number of monochromatic solutions to multiplicative equations

The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2, \dotsc, N\}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2} \log N$ monochromatic solutions, and that the leading constant is sharp. For $r=3$ and $r=4$ we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.