The Erdős Discrepancy Problem is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$. This was recently shown to be true by Tao (building on Polymath5).
Now consider the following stronger conjecture, which also generalizes van der Waerden.
In any $k$-coloring of the naturals for any $n$ there is a monochromatic sequence $(i+1)d, (i+2)d, \ldots (i+n)d$.
If true, this would of course be quite a strong result, so I more expect that someone might be able to show a simple counterexample to it. What about the even stronger density version?