Representations of matroids

The concept of matroids was originally introduced by Whitney and Van der Waerden in the 1930's to generalise the notion of linear dependence in a vector space; certain axioms satisfied by this relation were observed to be satisfied by other types of ’ dependence’ relations, such as algebraic dependence and ’ cycle’ dependence in a graph. Consequently a matroid was defined to be a set with an abstract dependence relation satisfying these axioms. One of the most natural questions to ask is whether every such ’ matroid' is representable in the obvious sense in a vector space. The answer is of course no (otherwise matroid theory would be equivalent to linear algebra) although in the early years of the subject examples of non-representable matroids were not easily obtainable. In this thesis we continue the work of Inglcton (in [20]) and Vamos (in [35,36]) on the representation problem, buiding up to an algebraic treatment in the important last chapter.