# Nearly Finitary Matroids

## TL;DR

This thesis explores nearly finitary matroids, introducing new concepts, analyzing their properties, and investigating their spectrum, with the aim of understanding their structure and the relationship between nearly finitary and k-nearly finitary matroids.

## Contribution

It introduces the notion of spectrum for nearly finitary matroids, constructs classes of such matroids, and explores their properties and generalizations, advancing the theoretical understanding of infinite matroids.

## Key findings

- Spectrum of nearly finitary matroids can be arbitrarily large.
- Finite rank matroids are unionable.
- Examples of nearly finitary independence systems that are not matroids.

## Abstract

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 $k$.