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Suppose that one is given \(v\) elements, and wishes to form a design that covers all \(t\)-sets from these elements exactly once. The design is to obey the further restriction that the longest block in the design has \(k\) 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 \(\text{g}^{(\text{k})}(1,t;v)\); 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 \(t = 2\) (pairwise balanced designs) and \(t = 3\) (designs with balance on triples). This survey paper presents the subject from its inception up to current results.