What is the following variation on Set Cover known as? What is the following variation on set cover known as?

Given a set S, a collection C of subsets of S and a positive integer K, do there exist K sets in C such that every pair of elements of S lies in one of the selected subsets.

Note: It is not hard to see that this problem is NP-Complete:  Given a normal set cover problem (S, C, K), make three copies of S, say S', S'', and S''',  then create your subsets as S''', |S| subsets of the form {a'} U {x in S'' | x != a} U {a'''}, |S| subsets of the form {a''} U {x in S' | x != a} U {a'''}, {a', a'' | a in C_i}.  Then we can solve the set cover problem with K subsets iff we can solve the pair cover problem with K + 1 + 2 |S| subsets.

This generalizes to triples, etc.  I would like to be able to not waste half a page proving this, and it is probably not obvious enough to dismiss as trivial.  It is certainly sufficiently useful that someone has proved it, but I have no idea who or where.

Also, is there a good place to look for NP-Completeness results that are not in Garey and Johnson?
To answer your second question, the Kahn-Crescenzi compendium of NP-hardness results is a valuable source for hardness results, and also covers many variants of core G&amp;J problems. The entry for set cover is a good example of this.