Monochromatic sums and products in N

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x + y, xy}.We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of N. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics.As a corollary of our main theorem we obtain partition regularity for new types of equations, such as x 2 -y 2 = z and x 2 + 2y 2 -3z 2 = w.