# Motivating Generalized Extremal Problems

## Document Type

Presentation Abstract

## Presentation Date

2-6-2023

## Abstract

In this talk, we discuss the history of extremal graph theory, in particular motivating several generalizations of classical extremal questions which have become popular in recent years. The foundational question of extremal graph theory is to determine the maximum number of edges in an *n *vertex graph which does not contain some "forbidden" graph *F *as a subgraph. After a century of work, this question has been well studied (though is still not well understood in some cases), revealing a rich underlying theory. But does the extremal question still make sense in a "colorful" graph setting? What about for hypergraphs? What if, instead of seeking to maximize the number of edges in an *F*-free graph, we wish to have an abundance of copies of some other subgraph *H*? We shall see that all of these directions naturally grow from the classical extremal question, and that we can often find "generalized" versions of "classical" theorems, e.g., supersaturation and stability results. However, we shall also highlight generalized phenomena which diverge from classical expectations. This talk assumes no background in extremal graph theory and is intended primarily to introduce history and motivation for generalized extremal questions.

## Recommended Citation

Halfpap, Anna, "Motivating Generalized Extremal Problems" (2023). *Colloquia of the Department of Mathematical Sciences*. 659.

https://scholarworks.umt.edu/mathcolloquia/659

## Additional Details

Link to the presenter's dissertation.

February 6, 2023 at 3:00 p.m. Math 103