Minimum number of sets of points
I can't access the paper itself, but judging by its abstract and a mention on p. 5 of this paper, a slight generalisation of your problem was proved NP-hard in 1978: Complete-Link Cluster Analysis by Graph Coloring (Pierre Hansen and Michel Delattre).  I say "slight generalisation" because (I assume that) the problem proved hard allows arbitrary distances between points; it's possible that Euclidean version of the problem, which you seem to have, is easier.  Maybe even the metric version is easier.

For the case of exactly 2 clusters, Wikipedia gives a poly-time algorithm based on 2SAT.