Golden-Mean and Secret Sharing Matroids

Maximum-sized results are an important part of matroid theory, and results currently exist for various classes of matroids. Archer conjectured that the maximum-sized golden-mean matroids fall into three distinct classes, as op- posed to the one class of all current results. We will prove a partial result that we hope will lead to a full proof. In the second part of this thesis, we look at secret sharing matroids, with a particular emphasis on the class of group-induced p-representable matroids, as introduced by Matúš. We give new proofs for results of Matúš', relating to M(K₄), F₇ and F⁻₇. We show that the techniques used do not extend in some natural ways, and pose some unanswered questions relating to the structure of secret sharing matroids.