Is there a relationship between the cost of a diamond and the cost of the ring setting?

Well no not really. The cost of a diamond is usually independent of the cost of a diamond ring setting. The only relationship I can think of between these two values is when a diamond is fancy cut, has an unusual shape or is very large in size. In that case, a run of the mill ring setting may not do, and you may need to have your ring setting custom made in order to accommodate an unusual diamond.