Date of Award

8-2025

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ping Zhang, Ph.D.

Second Advisor

Gary Chartrand, Ph.D.

Third Advisor

John Martino, Ph.D.

Fourth Advisor

Ebrahim Salehi, Ph.D.

Keywords

Edge-disjoint, monochromatic subgraphs, Ramsey number, red-blue edge coloring, vertex-disjoint

Abstract

Ramsey theory, though a relatively young branch of mathematics, has captivated the attention of graph theorists, combinatorialists, and theoretical computer scientists alike through its raw beauty, versatility, and powerful applications. Before it emerged as a branch of mathematics, the central idea of Ramsey theory appeared in the form of three lemmas in three separate papers by three different mathematicians working on three distinct areas of research. The first such lemma was published by David Hilbert in 1892, followed by the second lemma published by Issai Schur in 1916. However, Frank Ramsey’s renowned lemma, published in 1930, compelled mathematicians to establish its study shortly afterwards. In essence, Ramsey’s lemma, now called Ramsey’s theorem, shows that total disorder is impossible in large structures. Given a sufficiently large structure, some predefined configuration always appears. While the existence of a sufficiently large structure is guaranteed, the question “how large is sufficient?” remains. The minimum such number which answers this question is the quest of Ramsey theory and is aptly dubbed the Ramsey number. Specifically, in graph theory, given two graphs F and H, the Ramsey number R(F, H) is the smallest positive integer n such that for every red-blue coloring of the edges of Kn there is either a subgraph isomorphic to F , every edge of which is colored red, or a subgraph isomorphic to H, every edge of which is colored blue.

Studying Ramsey numbers leads to a natural extension. For a graph F and a positive integer t, the vertex-disjoint Ramsey number V Rt(F ) is the minimum positive integer n such that every red-blue coloring of the edges of Kn results in t pairwise vertex- disjoint monochromatic copies of subgraphs isomorphic to F, while the edge-disjoint Ramsey number ERt(F ) is the corresponding number for edge-disjoint subgraphs. Using this extension of Ramsey numbers, we explore and classify some families of Ramsey numbers in this research. Specifically, we investigate the vertex- and edge-disjoint Ramsey numbers of graphs of size 2 and 3 without isolated vertices, namely F ∈ {2K₂, P₃, P₄, P₃ + P₂, K₃, K₁,₃, 3K₂}. We find exact formulas for V Rt(F ) and calculate ERt(F ) for small values of t. When an explicit formula for ERt(F ) is not found, bounds are established. Additionally, we investigate the well-known class of stars K₁,n of order n + 1 and determine the vertex-disjoint Ramsey number for some specific n and t as well as explore the edge- disjoint Ramsey number for all values of n when t is small and develop bounds when t is large.

Access Setting

Dissertation-Open Access

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