Van der Waerden type numbers (for geometric progressions)

Question

Van der Waerden theorem is true also for geometric progressions. Is there anything interesting in van der Waerden type numbers $ W'(r,k) $ derived from this version? ($ W'(r,k) $ is such that if the integers $ \{1, 2, ..., W'(r,k)\} $ are colored, each with one of r different colors, then there are at least k integers in geometric progression all of the same color). Probably there is some obvious connections between $ W'(r,k) $ and $ W(r,k) $. Is there any literature about this topic?