Abstract
Publisher Summary This chapter focuses on triple systems, block designs, and quadruple systems. The chapter provides a comprehensive survey of a large subset of the present day state of the art. Christine Treash's result that a partial triple system can always be embedded in a triple system is the genesis of the study of embedding theorems for block designs. Although Treash's Theorem has subsequently been improved upon with respect to the size of the containing triple system, the chapter presents Treash's original proof. There are two reasons for this: The first is the historical importance of the result, and the second is the motivation for the techniques used later by B. Ganter and R. W. Quackenbush to embed partial block designs with block sizes larger than 3.
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