TITLE: Is this Ramsey-type problem an open problem?
QUESTION [9 upvotes]: A blog claims that the following Ramsey-type (or van der Waerden type) problem is open:
If the natural numbers are colored with finitely many colors, must there exist x and y (not both 2) such that x+y and xy are the same color?
Is it correct that this is an open problem, and can anybody help me track down a reference?

REPLY [17 votes]: Joel Moreira has now proved that we always get monochromatic sets of the shape $\{x, xy, x+y\}$.  The main result is much more general and uses a connection with topological dynamics, but he includes a beautifully simple direct proof of this special case, assuming only an easy to check consequence of van der Waerden's theorem.