## Graduate Theses, Dissertations, and Problem Reports

### Infinite Matroids and Transfinite Sequences

Summer

2019

Dissertation

PhD

#### College

Eberly College of Arts and Sciences

Mathematics

#### Committee Chair

Jerzy Wojciechowski

#### Committee Co-Chair

Krzysztof Ciesielski

#### Committee Member

Krzysztof Ciesielski

Hong-Jian Lai

John Goldwasser

Elaine Eschen

#### Abstract

A matroid is a pair M = (E,I) where E is a set and I is a set of subsets of E that are called independent, echoing the notion of linear independence. One of the leading open problems in infinite matroid theory is the Matroid Intersection Conjecture by Nash-Williams which is a generalization of Hall’s Theorem. In [31] Jerzy Wojciechowski introduced µ- admissibility for pairs of matroids on the same ground set and showed that it is a necessary condition for the existence of a matching. A pair of matroids (M,W) with common ground set E is µ-admissible if a subset of sequences in E × {0,1} have a certain property. In order to determine if this property implies anything about the length of the sequence, we modify µ-admissibility to obtain µ 0 -admissibility, an equivalent property for pairs of matroids. We then use µ 0 -admissibility to show that for every successor ordinal of the form α + 2n there is a pair of partition matroids such that a shortest sequence in E × {0,1} that fails to have the desired property has length α + 2n. Furthermore, we introduce the class of patchwork matroids, which contains all finite matroids and all uniform matroids, and provide a method for their construction, prove a characterization theorem, show the class is closed under duality and taking minors, as well as several other properties. Lastly, a cyclic flat is a flat which is a union of circuits. We combine this notion with that of trees of matroids, which are introduced in [7] to show that the lattice of cyclic flats of a locally finite tree of finite matroids contains an atomic element.

COinS

#### DOI

https://doi.org/10.33915/etd.4121