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.
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
2) N Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227-245.
(This contains for example Graham's proof and extensions)
3) R.K. Guy, Unsolved problems in number theory, section E29