On the number of monochromatic solutions to multiplicative equations

The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2,\dots ,N\}$, 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 $(1\mathrm{/}2\sqrt{2})N^{1\mathrm{/}2}\log N$ 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.