Topological cycle matroids of infinite graphs

We prove that the topological cycles of an arbitrary infinite graph together with its topological ends form a matroid. This matroid is, in general, neither finitary nor cofinitary.Emmanuel Colleg

The structure of 2-separations of infinite matroids

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.Comment: 31 page

All graphs have tree-decompositions displaying their topological ends

We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992

Nearly Finitary Matroids

In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system for this thesis. In Chapter 2, we summarize our main results after reviewing historical background and motivation. In Chapter 3, we define a notion of spectrum for matroids. Moreover, we show that the spectrum of a nearly finitary matroid can be larger than any fixed finite size. We also give an example of a matroid with infinitely large spectrum that is not nearly finitary. Assuming the existence of a single matroid that is nearly finitary but not $k$-nearly finitary, we construct classes of matroids that are nearly finitary but not $k$-nearly finitary. We also show that finite rank matroids are unionable. In Chapter 4, we will introduce a notion of near finitarization. We also give an example of a nearly finitary independence system that is not $k$-nearly finitary. This independence system is not a matroid. In Chapter 5, we will talk about Psi-matroids and introduce a possible generalization. Moreover, we study these new matroids to search for an example of a nearly finitary matroid that is not $k$-nearly finitary. We have not yet found such an example. In Chapter 6, we will discuss thin sums matroids and consider our problem restricted to this class of matroids. Our results are motivated by the open problem concerning whether every nearly finitary matroid is $k$-nearly finitary for some $k$.Comment: PhD Thesis, UC Davis (2018