Research & Projects
Research & Projects
I completed this project with collaborators Joanna Boyland (Harvard), Bill Gasarch (UMD), and Nathan Hurtig (Rose-Hulman) as a part of the REU-CAAR program at the Department of Computer Science at thee University of Maryland in 2023.
Abstract: Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let S be a linearly ordered set and a∈ N. The big Ramsey degree of a in S, denoted T(a,S), is the least integer t such that, for any finite coloring of the a-subsets of S, there exists S′⊆ S such that (i) S′ is order-equivalent to S, and (ii) if the coloring is restricted to the a-subsets of S′ then at most t colors are used.
Mašulović \& Šobot (2019) showed that T(a,ω+ω)=2a. From this one can obtain T(a,ζ)=2a. We give a direct proof that T(a,ζ)=2a.
Mašulović and Šobot (2019) also showed that for all countable ordinals α<ω^ω, and for all a∈ N, T(a,α) is finite. We find exact value of T(a,α) for all ordinals less than ω^ω and all a∈ N.
Joint Mathematics Meetings; January 2023; Boston, MA
BUGCAT Conference; November 2023; Binghamton, NY
Currently Under Review with Combinatorica
This work was done my senior year at TCNJ under the supervision of Dr. Andrew Clifford as my final project for my Honors Thesis.
Abstract: The Whitehead Model of free groups can be used to measure the complexity, or degree, of automorphisms of free groups. The bound for the degree of the f∘ g for deg(f)= deg(g)=0 had previously been discovered. We extend this result to the case where at least one of our automorphisms has degree 0.
Senior Honors Thesis; May 2023; Ewing, NJ