Is this Ramsey-type problem an open problem? 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?
 A: Almost this problem appears as Question 11 in Vitaly Bergelson's 1996 paper
Ergodic Ramsey Theory - An Update. (In that paper he asks whether $x$, $y$, $x+y$ and $xy$ can all have the same colour.
A: 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.
A: 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
A: 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
