  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 $\mathbb{N}$. 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 $x^2+2y^2-3z^2=w$.
