Date of Award
Doctor of Philosophy
Dr. Andrzej Dudek
Dr. Patrick Bennett
Dr. Alan Frieze
Dr. Allen Schwenk
Graph theory as a mathematical branch has been studied rigorously for almost three centuries. In the past century, many new branches of graph theory have been proposed. One important branch of graph theory involves the study of extremal graph theory. In 1941, Turán studied one of the first extremal problems, namely trying to maximize the number of edges over all graphs which avoid having certain structures. Since then, a large body of work has been created in the study of similar problems. In this dissertation, a few different extremal problems are studied, but for hypergraphs rather than graphs. In particular, we consider the saturation problem for families of Berge hypergraphs.
In addition to the study of extremal hypergraph problems, this dissertation also focuses on problems of a probabilistic nature. Graphs and hypergraphs that are generated through a random process have been popular objects of study since the seminal work on them done by Erdős and Rényi, and independently gilbert in 1959. Here we study problems involving finding certain Ramsey properties of random hypergraphs, and also finding the probabilistic threshold for specific cycle structures to appear in randomly colored random Hypergraphs. We also explore an application of Markov chains to a problem of a topological nature, namely studying Morse functions on the sphere.
English, Sean, "Probabilistic and Extremal Problems in Combinatorics" (2018). Dissertations. 3319.