This problem appears on page $134$ of Peter Winkler's Mathematical Mind-Benders.
Can you color the natural numbers $\{0,1,2,\dots\}$ with finitely many colors, in such a way that the sum $x+y$ and the product $xy$ of any two whole numbers always have different colors?
This was unsolved at the time of writing. Winkler says,
Apparently a set of six numbers is known such that any two of them are the product and sum of some pair of numbers, so that at least six colors would be needed; on the other hand, there are not arbitrarily large sets of numbers with the above property.
I haven't been able to find six such numbers, though I admit I haven't tried very hard, but it's the statement that it is known that there are not arbitrarily large cliques that intrigues me. I can't imagine how to attack a question like this.
I haven't been able to find anything about this on the Web. Can you give me any information about the problem, and how to produce the results mentioned above?