Almost Varieties and Extremal Matroids

Abstract

This thesis, as a body of work, examines minor-closed classes of matroids that satisfy a weaker definition than varieties (introduced by Kahn and Kung [9]). These new classes are called almost varieties. To assist in the description of almost varieties, we give some basic properties of a particular type of infinite matroid, called a locally-finite matroid. We begin to classify the almost-varieties of low-connectivity and show they can be partitioned into classes that are similar to varieties with low connectivity. The main theorems of this thesis are constructions that produce new almost varieties from old. One reason for studying these minor-closed classes is that, for collections of matroids representable over certain partial fields, the locally finite matroid associated with a particular almost-variety will model the maximum sized matroids of the class. The end of this thesis is a study of all the rank-3 and rank-4 matroids representable over the golden-mean partial-field.