Is this Ramsey-type problem an open problem?

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.

The problem (and several extensions) was mentioned by Hindman.

The case of two colours was solved by Graham:
The interval $[1,252]$ contains $x$ and $y$ such that $x,y, x+y$ and $xy$ are all monochromatic, and 252 is minimal.

References:

1) J Fox, Yeu-Whai Kathy Lin, and M Thibaul
The Clique Number of the Graph of Pairwise Sums and Products is 3 or 4

http://math.mit.edu/classes/18.821/documents/sample.pdf

2) N Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227-245.

http://www.ams.org/journals/tran/1979-247-00/S0002-9947-1979-0517693-4/home.html

(This contains for example Graham's proof and extensions)

3) R.K. Guy, Unsolved problems in number theory, section E29

The question you pose is (currently) still open, but there was an interesting result by my friend Peter Blanchard which proves a "divisible" version of the problem. Namely, given a finite coloring, there are $x$, $y$ and $x+y$ all with the same color, and $x|y$.

Pseudo-arithmetic sets and Ramsey theory