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



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> 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.
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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!




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**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.



Combining van der Waerden's theorem with Ramsey's theorem
The way I understand the question is that $k$-tuples of integers are considered so that each coordinate lies in $[1,N]$, so in other words one considers $[1,N]^k$ and colors all the elements of this set, so each poin $(a1,...,ak)$ gets some color. 


There is a version of Szemerédi's Theorem for $\mathbb{Z}^k$, due to Furstenberg and Katznelson, that asserts that for every subset $A$ of $\mathbb{Z}^k$ of positive upper Banach density and any finite $F\subset \mathbb{Z}^k$ there is some $u \in \mathbb{Z}^k$ and some $n \in \mathbb{N}$ such that $u+nF \subset A$. 


For this result see for example Corollary at the start of [the paper by Bergelson and Leibman](http://www.math.osu.edu/~bergelson.1/PolSz.pdf), establishing a polynomial version of Szemerédi's theorem (I failed to properly locate the original paper, thus this reference by proxy). 


This result necessarily also has a 'finitary' analogue (to match the question), however the proof being "ergodic" to get explict constants could be a problem.
Also, as usual, on would consider for a given coloring the largest set of the same color, which necessarily has positive upper denisty. 


Now, this result implies directly the existence of s-term arithmetic progression in such a set $A$ in the sense I understand this word, namely as a set of the form $a, a+d, ..., a+(s-1)d$ with $a,d$ in the relevant structure so in this case $\mathbb{Z}^k$, by taking $F$ some set of points in arithmetic progression. 


However, this appears (after the update) not to be what is meant (yet I have to admit in retrospect that my argument for my, and I think the common, use of AP seems overly complicated).


Yet what OP is looking for is an arithemtic progression $S$ in $\mathbb{Z}$ (or rather $[1,N]$) such that $S^k$ is monochromatic. As noted by OP what the mentioned result gives is a monochromatic set of the form $u+S^k$. And, as mentioned in comments to actually get $S^k$ monochromatic is impossible. 


(Sorry for the repeated edits.)


