Consider positive integers $c$, $k$, and $s$. Does there exist some $N = N(c,k,s)$ such that the following holds?

Take any $c$-coloring of the $k$-tuples of integers in $[1,N]$. Then there is an arithmetic progression of length $s$ such that all $k$-tuples of it have the same color. More precisely, there exists a set $S \subseteq [1,N]$ such that $|S| = s$, $S$ is an arithmetic progression, and all tuples in $S^k$ have the same color.

If $k=1$, then this is van der Waerden's theorem. If I don't care for an arithmetic progression (and am happy with just a set), then this is Ramsey's theorem. This is asking for the best of both theorems.

I was unable to find such a statement in my searches. I thought this might be a consequence of the Hales-Jewett Theorem, but could not show this. It seems like a first principles approach combining proofs of van der Waerden and Ramsey's theorem might work. But I was wondering if this is already known.

Thanks!

**Update**

User Wei Wang has refuted this statement below with a simple counterexample. Exampled added here for clarity. Suppose $N(2,3,4)$ existed. Simply color all three tuples in $[1,N]$ that form an AP blue, and remaining tuples red. Consider any AP $S = (s_1, s_2, s_3, s_4)$. The tuple $\{s_1, s_2, s_3\}$ is blue and $\{s_1, s_3, s_4\}$ is red.

User quid in his answer below has given a reference showing that a variant of my question is true. If I understand correctly, it is the case that there are k APs $S_1, S_2, \ldots, S_k$ such that tuples in $S_1 \times S_2 \times \cdots \times S_k$ have the same color. One can also ensure that these APs have the same minimum difference. (One can get a lot more, as explained in quid's post.) All in all, I think his post gives "more than an answer" to my question.