Let $P(n)$ be the statement "any $n$ coloring of $\mathbb{N}$ contains a monochromatic progression $a, a+d, a+2d$ such that $d>a$".

For which $n$ is $P(n)$ true?

It's easy to see that $P(2)$ is true by a simple modification of the color focusing argument that is used in the traditional proof of van der Waerden's theorem. However, this argument does not seem to generalize to more colors, or at least not very easily.

It's also easy to see that a similar statement is not true for progressions of length $4$, even in the $2$ color case: just color $[2^n,2^{n+1}-1]$ red if $n$ is even and blue if $n$ is odd. If $2^n<d<2^{n+1}$ then $a+d$ or $a+2d$ is in $2^{n+1}$ but $a+2d$ or $a+3d$ is in $2^{n+2}$.

I asked the same question on math stack exchange a little over a week ago but got no replies or comments, so I figured I'd ask here as well.