#### Date of Award

1-2021

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematics

#### Thesis Sponsor/Dissertation Chair/Project Chair

Deepak Bal

#### Committee Member

Aihua Li

#### Committee Member

Jonathan Cutler

#### Abstract

The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn).

#### File Format

#### Recommended Citation

Schudrich, Eliyahu, "Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths" (2021). *Theses, Dissertations and Culminating Projects*. 684.

https://digitalcommons.montclair.edu/etd/684