A question that received a lot of attention over the last years are quantitative refinements of Szemerédi's theorem, a generalisation of van der Waerden's Theorem asserting that fixing any positive density $d$ one will find arithemetic progressions in a set of that density provided the initial segment of integers one considers is large enough. (If one has $c$ colors one is certainly guaranteed a set of density $1/c$ with all elements the same color.)

Now, it is known that one even can let the density go to zero (sufficiently slowly depending on the size of the initial segment). How slowly is not quite clear, and this is what these quantitiative refinemenets are about. See also for a conjecture regarding this.

For (recent) contributions on this see, e.g.,

a paper giving much better bounds in general ($k$ arbitrary) than known before, by Gowers for the general case

the to my knowledge currently best bounds for $k=4$ by Green and Tao

the to my knowledge currently best bounds for $k=3$ by Sanders

A feature these three papers share is that they take a Fourier analytic approach to the problem. And for this it *is* advantageous to work over a finite (prime) cyclic group rather then over the integers. Note that while all three papers phrase the main relsult for integers, for the proof they actually pass to a finite (prime) cyclic group.

I am not sure this answer is exactly what you are looking for, as while the problem is in some sense essentially the same it might (or might not) be the case that you are looking for results of a somewhat different flavor (say, less 'asymptotic').

In any case, working over a prime cyclic group certainly has some advantage and the problem in this context was considered *a lot*; indeed many of the recent works in this context even if at first glance for integers in the end *are* results for *(prime) cyclic groups*. Yet, there is not a huge difference in the bounds in the end for the reason Anthony Quas mentioned.