Van der Waerden's Theorem
Given positive integers $r$ and $k$, there is some number $N$ such that if the integers $\{1, 2, \dots, N\}$ are colored, each with one of $r$ different colors, then there are at least $k$ integers in arithmetic progression whose elements are of the same color.
This is often described as a ''Ramsey-like'' theorem, i.e., a theorem similar to Ramsey's Theorem.
Ramsey's Theorem
Given positive integers $r$ and $k$, there is some number $N$ such that if the vertices of the complete graph $K_N$ are colored, each with one of $r$ different colors, then there is a monochromatic $k-$clique.
The similarities are obvious: Both theorems are about unavoidable structures in colored sets. Ramsey's Theorem gets to name this class of results since a great portion of them is a consequence of it. However, I've never seen a proof of Van der Waerden's Theorem through Ramsey's Theorem. I wonder if there is such.
A stronger version of Ramsey's Theorem might be useful.
Ramsey's Theorem
Given positive integers $r$, $k$ and $n$, there is some number $N$ such that if the subsets with $n$ elements of $\{1, 2, \dots, N\}$ are colored, each with one of $r$ different colors, then there is an homogeneous set of size $k$.
Here, an homogeneous set is a set such that every subset with $n$ elements has the same color. Notice the first version is the case $n = 2$.
In particular, I would like to know the following:
Given positive integers $r,~k,~n$ and $N$ such that the integers $\{1, 2, \dots, N\}$ are colored, each with one of $r$ different colors, is there a way to color the subsets with $n$ elements of $\{1, 2, \dots, N\}$ is such a way so that an homogeneous set of size $k$ is a monochromatic arithmetic progression?
The monochromatic bit is easy to ensure. If fact, notice that it's enough to prove the statement for big enough values of $k$. So choose $k > rn$ and color a subset white if it has elements with different colors. The Pigeonhole Principle tell us that there no white homogeneous set of size $k$ and therefore any homogeneous set is monochromatic. Now, can the coloring force an arithmetic progression?