Matroid theory is a vibrant area of research that provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to any mathematician interested in the geometric approach to matroids. Written in a friendly, fun-to-read style and developed from the authors' own undergraduate courses, the book is ideal for students. Beginning with a basic introduction to matroids, the book quickly familiarizes the reader with the breadth of the subject, and specific examples are used to illustrate the theory and to help students see matroids as more than just generalizations of graphs. Over 300 exercises are included, with many hints and solutions so students can test their understanding of the materials covered. The authors have also included several projects and open-ended research problems for independent study.

A tight upper bound on the number of elements in a connected matroid with fixed rank and largest cocircuit size is given. This upper bound is used to show that a connected matroid with at least thirteen elements contains either a circuit or a cocircuit with at least six elements. In the language of matroid Ramsey numbers, n(6, 6) = 13: this is the largest currently known matroid Ramsey number.

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In this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently, of Seymour. The following consequence of the main result generalizes a graph result of Thomassen and Toft on induced non-separating cycles and another graph result of Kaugars on deletable vertices. If G is a simple 2-connected graph with minimum degree at least 3, then, for every edge e, there are at least two induced non-separating cycles avoiding e and two deletable vertices non-incident to e. Moreover, G has at least four induced non-separating cycles. Academic Press