Browsing Theses by Supervisor "Nelson, Peter"
Now showing items 1-7 of 7
-
Clique minors in dense matroids
(University of Waterloo, 2022-09-23)The objective of this thesis is to bound the number of points a $U_{2,\ell+2}$- and $M(K_{k+1})$-minor-free matroid has. We first prove that a sufficiently large matroid will contain a structure called a tower. We then use ... -
Counting Pentagons in Triangle-free Binary Matroids
(University of Waterloo, 2020-07-30)A rank-n binary matroid is a spanning subset E of F₂ⁿ\{0}, a triangle is a set of three elements from E which sum to zero, and the density of a rank-n binary matroid is |E|/2ⁿ. We begin by giving a new exposition of a ... -
Enumerating matroid extensions
(University of Waterloo, 2023-09-01)This thesis investigates the problem of enumerating the extensions of certain matroids. A matroid M is an extension of a matroid N if M delete e is equal to N for some element e of M. Similarly, a matroid M is a coextension ... -
Formalizing the Excluded Minor Characterization of Binary Matroids in the Lean Theorem Prover
(University of Waterloo, 2024-01-23)A matroid is a mathematical object that generalizes the notion of linear independence of a set of vectors to an abstract independence of sets, with applications to optimization, linear algebra, graph theory, and algebraic ... -
Induced Binary Submatroids
(University of Waterloo, 2021-07-21)The notion of induced subgraphs is extensively studied in graph theory. An example is the famous Gy\'{a}rf\'{a}s-Sumner conjecture, which asserts that given a tree $T$ and a clique $K$, there exists a constant $c$ such ... -
On The Density of Binary Matroids Without a Given Minor
(University of Waterloo, 2016-12-21)This thesis is motivated by the following question: how many elements can a simple binary matroid with no $\PG(t,2)$-minor have? This is a natural analogue of questions asked about the density of graphs in minor-closed ... -
Quadratically Dense Matroids
(University of Waterloo, 2020-07-08)This thesis is concerned with finding the maximum density of rank-$n$ matroids in a minor-closed class. The extremal function of a non-empty minor-closed class $\mathcal M$ of matroids which excludes a rank-2 uniform ...