In this thesis we first investigate whether all finite partial linear spaces are embedded in finite projective planes, a question posed by Erdös. We obtain the partial result that every finite partial linear space is embedded in a translation net generated by a partial spread and discuss future avenues of research which may lead to a solution. We also investigate the structure of the Erdös-Rényi Graph. In particular, we construct two families of symmettric graphs of girth 5 which are induced subgraphs of the Erdös-Rényi Graph. We also consider the question of whether a subgraph of the Erdös-Rényi Graph is an extremal C₄-free graph, and obtain the partial result that a counterexample must have degree equal to q + 2, where q is a prime power of 2. Lastly, we investigate the independence number of the Erdös-Rényi Graph, and find magnitude by eigenvalue methods and explicit constructions of independent sets.