#### 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. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H_{1}, ..., H_{k}, we write G → (H_{1}, ..., H_{k}), or G → (H)_{k} when H_{1} = ··· = H_{k} = H, if every k-edge-coloring of G contains a monochromatic H_{i} in color i for some i ∈ {1,...,k}. The Ramsey number r_{k}(H_{1}, ..., H_{k}) is the minimum integer n such that K_{n} → (H_{1}, ..., H_{k}), where K_{n} is the complete graph on n vertices. Computing r_{k}(H_{1}, ..., H_{k}) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai colorings, that is, edge-colorings with no rainbow triangles. From this we define the Gallai-Ramsey number gr_{k}(K_{3},G) as the minimum integer n such that either K_{n} contains a rainbow triangle, or K_{n} → (G)_{k} . In this thesis, we determine the Gallai-Ramsey numbers for C_{7} with multiple colors. We believe the method we developed can be applied to find gr_{k}(K_{3}, C_{2n+1}) for any integer n ≥ 2, where C_{2n+1} denotes a cycle on 2n + 1 vertices.

#### Thesis Completion

2017

#### Semester

Spring

#### Thesis Chair/Advisor

Song, Zi-Xia

#### Degree

Bachelor of Science (B.S.)

#### College

College of Sciences

#### Department

Mathematics

#### Degree Program

Mathematics

#### Location

Orlando (Main) Campus

#### Language

English

#### Access Status

Open Access

#### Release Date

11-1-2017

#### Recommended Citation

Bruce, Dylan, "Gallai-Ramsey Numbers for C7 with Multiple Colors" (2017). *Honors Undergraduate Theses*. 264.

https://stars.library.ucf.edu/honorstheses/264