Victor Souza

On the number of monochromatic solutions to multiplicative equations

Given an r-colouring of the interval {2,...,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 (1/2sqrt(2)) N^(1/2) log(N) monochromatic solutions, and that the leading constant is sharp. We also establish a stability version of this result. For general r, we show that there are at least C_r N^(1/S(r-1)) monochromatic solutions, where S(r) is the Schur number for r colours and C_r is a constant. This bound is sharp up to logarithmic factors when r <= 4. We also obtain results for more general multiplicative equations. Based on joint work with Lucas Aragão, Jonathan Chapman and Miquel Ortega.