#### Department

Mathematics

#### First Advisor

Bryan Shader

#### Description

In a broad sense, graph theory has always been present in civilization. Graph theory is the math of connections β at a party, who knows each other? How many handshakes will each person have to give before shaking hands with everyone? What is the best way to drive from city to city? Extremal graph theory is a branch that deals with counting items (called vertices) and connections between two items (called edges) and determining the maximum/minimum number of characteristics needed to satisfy a certain property. The specific topic of this talk is TurΓ‘n numbers, a topic of extremal graph theory that attempts to determine the maximum number of edges a graph may have without a specified pattern emerging. For two graphs, πΊπΊ and π»π», the TurΓ‘n number is denoted ππππ(πΊπΊ,π»π»), and is the maximum number of edges in a subraph of πΊπΊ that contains no copy of π»π». We were able to find and prove the previously unknown TurΓ‘n number for a certain pattern in a certain graph. To be precise, we found the TurΓ‘n number of ππ copies of vertex-disjoint cliques in ππ-partite graphs (part sizes ππ1,β¦ππππ). That is,

ex(Kn_{1},n_{2},...,n_{r}, kKr) = β n_{i}n_{j }β n_{1}n_{2 }+ n_{2}(kβ1)

where the sum goes from 1β€ i < jβ€ r.

This talk will describe the motivation and history of extremal graph theory, discuss definitions and concepts related to the research that was done, explain the main concept behind the proof, and finally discuss possible future research.

TurΓ‘n Numbers of Vertex-disjoint Cliques in r-Partite Graphs

In a broad sense, graph theory has always been present in civilization. Graph theory is the math of connections β at a party, who knows each other? How many handshakes will each person have to give before shaking hands with everyone? What is the best way to drive from city to city? Extremal graph theory is a branch that deals with counting items (called vertices) and connections between two items (called edges) and determining the maximum/minimum number of characteristics needed to satisfy a certain property. The specific topic of this talk is TurΓ‘n numbers, a topic of extremal graph theory that attempts to determine the maximum number of edges a graph may have without a specified pattern emerging. For two graphs, πΊπΊ and π»π», the TurΓ‘n number is denoted ππππ(πΊπΊ,π»π»), and is the maximum number of edges in a subraph of πΊπΊ that contains no copy of π»π». We were able to find and prove the previously unknown TurΓ‘n number for a certain pattern in a certain graph. To be precise, we found the TurΓ‘n number of ππ copies of vertex-disjoint cliques in ππ-partite graphs (part sizes ππ1,β¦ππππ). That is,

ex(Kn_{1},n_{2},...,n_{r}, kKr) = β n_{i}n_{j }β n_{1}n_{2 }+ n_{2}(kβ1)

where the sum goes from 1β€ i < jβ€ r.

This talk will describe the motivation and history of extremal graph theory, discuss definitions and concepts related to the research that was done, explain the main concept behind the proof, and finally discuss possible future research.

## Comments

Honors Program

Research partners: Jessica De Silva, University of Nebraska-Lincoln.Kristin Heysse, Iowa State University. Adam Kapilow, Swarthmore College. Michael Young, University of Wyoming.