Suppose we $r$ color $\mathbb{N}$. For what values of $r$ and $k$ are we guaranteed to find a monochromatic $k$ term arithmetic progression $a,a+d,...,a+(k-1)d$ with the additional property that $d>a$?

We can always find one in the $r=2,k=3$ case by a slight variation of the normal color focusing argument, but this does not seem to generalize to greater values in either of the parameters.

It's also easy to see that this will no longer be true for larger values of $k$ even when $r=2$: 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 $2^{n+2}$.

So the real question is for what values of $r$ does the $k=3$ version of this statement hold? I've tried adapting the counter example coloring I gave in the last paragraph to the $r=3$ setting but haven't been able to get it to work.


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