Abstract 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 (1/22) N^1/2 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.
Aragao et al. (Mon,) studied this question.