Van der Waerden’s theorem gives us a finite number $W(k,r)$ defined as the smallest positive integer $N$ such that for any $ngeq N$, any $r$-coloring of $(n)={1,dots,n}$ admits a monochromatic $k$-AP. We can ask the same question except with $(n)$ replaced by $mathbb{Z}/nmathbb{Z}$, calling the answer the “cyclic van der Waerden number” and denoting it by $W_c(k,r)$ (seems to be first mentioned in Burkert and Johnson, 2011). An immediate bound is that $W_c(k,r)leq W(k,r)$, so we know that $W_c$ is finite.

Is there any progress on determining the values of $W_c(k,r)$ that is not just “check every number not greater than $W(k,r)$“? Even if the exact values are not known for larger $k$ and $r$, are there any improved asymptotics on $W_c$ that are better than the Gowers bound for $W$? My quick literature search seems to only produce the Burkert and Johnson paper and a single other one (Grier, 2012) which computes $W_c(3,2)$ but nothing else.