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

likean chromatic number problem -- itisasking whether the chromatic number of a certain graph is infinite. (Of course, you must be aware of this or you wouldn't have posted your comment.) Unfortunately, that reformulation doesn't seem to help much, essentially because the graph is too concrete for general facts about graph colouring to be of any use. However, I did once attend a talk that discussed the problem in those terms. $\endgroup$