Completion and Embedding Problems for Combinatorial Designs

Combinatorial design theory studies arrangements and combinations of discrete objects according to different rules. Applications of designs are not only limited to analysis of experiments, but also useful in network analysis, cryptography and communication protocols, error correcting codes, mathematical biology, algorithm design, tournament scheduling, lotteries, etc. The topic of when a partial combinatorial design can be completed or embedded has attracted a great deal of interest over the years. In this thesis, we investigate four topics related to the completion or embedding of partial H-designs. We make progress on partial Steiner triple systems, partial block designs and partial star designs.