Suppose that one is given elements, and wishes to form a design that covers all -sets from these elements exactly once. The design is to obey the further restriction that the longest block in the design has elements in it; furthermore, we wish the design to contain as few blocks as possible.
The number of blocks in such a minimal design is denoted by the symbol ; determination of this number is closely connected with the behaviour of Steiner Systems. Recently, much progress has been made in two important cases, namely, when (pairwise balanced designs) and (designs with balance on triples). This survey paper presents the subject from its inception up to current results.
