#### Abstract

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For a graph H, the k-color Ramsey number r(H; k) of H is the smallest integer n such that every k-edge-coloring of K_{n} contains a monochromatic copy of H. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k is at least 3, also known as the multicolor Ramsey number of H. Let S_{n} denote the star on n+1 vertices, the graph with one vertex of degree n (the center of S_{n}) and n vertices of degree 1. The double star S(n,m) is the graph consisting of the disjoint union of S_{n} and S_{m} together with an edge joining their centers. In this thesis, we study the multicolor Ramsey number of double stars. We obtain upper and lower bounds for r(S(n,m); k) when k is at least 3 and prove that r(S(n,m); k) = nk + m + 2 for k odd and n sufficiently large. We also investigate a new variant of the Ramsey number known as the list Ramsey number. Let L be an assignment of k-element subsets of the positive integers to the edges of K_{n}. A k-edge-coloring c of K_{n} is an L-coloring if c(e) belongs to L(e) for each edge e of K_{n}. The list Ramsey number r_{l}(H; k) of H is the smallest integer n such that there is some L for which every L-coloring of K_{n} contains a monochromatic copy of H. In this thesis, we study r_{l}(S(1,1); p) and r_{l}(S_{n}; p)_{,} where p is an odd prime number.

#### Thesis Completion

2022

#### Semester

Spring

#### Thesis Chair/Advisor

Song, Zi-Xia

#### Degree

Bachelor of Science (B.S.)

#### College

College of Sciences

#### Department

Mathematics

#### Degree Program

Mathematics

#### Language

English

#### Access Status

Open Access

#### Release Date

5-1-2022

#### Recommended Citation

Ruotolo, Jake, "Multicolor Ramsey and List Ramsey Numbers for Double Stars" (2022). *Honors Undergraduate Theses*. 1197.

https://stars.library.ucf.edu/honorstheses/1197