# co.combinatorics – Computation of cyclic van der Waerden numbers

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.