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?



Is this Ramsey-type problem an open problem?
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


