Homogeneous van der Waerden

Question

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?

Answer 1

This is false already for $k=2,n=4$. Color an integer $m$ according to the parity of the exponent of $2$ in the prime factorization. Among $i+1,i+2,i+3,i+4$ at least one number is odd, and at least one is divisible by $2$ and not by $4$. Those two numbers have the parity of the exponent different, hence so do the corresponding two among $(i+1)d,(i+2)d,(i+3)d,(i+4)d$. Hence those four numbers can't have the same color.

Edit: for completeness, it's false for $k=2,n=3$ as well. For integer $m$, write it as $3^i\cdot j,3\nmid j$ and color $m$ according to $j\mod 3$. Then among $i+1,i+2,i+3$ the two which are not divisible by $3$ will have different remainders, do multiplying them by $d$ gives numbers of different colors. Hence your conjecture only holds in trivial cases $k=1$ and $n\leq 2$.