Matroids: Introduction

Abstract

Matroids were introduced in the early 1930’s in an attempt to axiomatize and generalize basic notions in linear algebra such as dependence, basis and span. The importance of matroids came to be appreciated with the discovery of new classes of matroids so that today we may rightly consider them as a unifying concept for a large part of combinatorics opening up basic combinatorial questions to algebraic ideas and methods. One of these fields is graph theory; in fact, it was precisely this correspondence between concepts in linear algebra and concepts in graph theory which set the theory of matroids on its way. Since then, other branches of combinatorics such as transversal theory, incidence structures and combinatorial lattice theory have been brought successfully into the realm of matroid theory. Indeed, it is this exchange of ideas from various fields which is one of the most gratifying aspects of matroid theory and also one measure of its success.