On the number of monochromatic solutions to multiplicative equations

The following question was asked by Prendiville: given an r -colouring of the interval \2, , N\ { 2 , ⋯ , N } , what is the minimum number of monochromatic solutions of the equation $$xy = z$$ x y = z ? For $$r=2$$ r = 2 , we show that there are always asymptotically at least (1/2√2) N1/2 log N ( 1 / 2 2 ) N 1 / 2 log N monochromatic solutions, and that the leading constant is sharp. For $$r=3$$ r = 3 and $$r=4$$ r = 4 we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations.