### Karen Meagher

Title: A Brief Introduction to World of Erd\H{o}s-Ko-Rado Theorems

Abstract: The Erd\H{o}s-Ko-Rado (EKR) theorem is a famous result that is one of the cornerstones of extremal set theory. This theorem answers the question “What is the largest family of intersecting sets, of a fixed size, from a base set?”

This result has been the starting point for a whole field of study. For example, this question may be asked for any type of object for which there is some notion of intersection or difference. There natural versions of the EKR theorem for permutations, vector spaces, designs, partial geometries, integer sequences, domino tilings, partitions, matchings and many other objects. There are also many related questions about the largest sets with additional restrictions.
There are many vastly different proofs of these results, some are simple straight-forward counting arguments, others use graph theory and some require nuanced properties of related matrix algebra. I will show some of the different proof methods for EKR theorems, with a focus on the best method, which is of course, the algebraic method.

The talk is available here, and you can view the slides here.

This talk was given on June 28th, 2022

Title: Group Theory and the Erd\H{o}s-Ko-Rado (EKR) Theorem

Abstract: Group theory can be a key tool in sovling problems in combinatorics; it can provide a clean and effective proofs, and it can give deeper understanding of why certain combinatorial results hold. My research has focused on the famous Erd\H{o}s-Ko-Rado (EKR) theorem. There are many proofs, extensions and generalization of this result. My favourite proofs are the ones that make use of finite groups, particularly automorphism groups. In this talk I will give some background on the EKR theorem and show some of the extensions and generalizations of this theorem. I will go over some of the different proofs, comparing the older methods with the newer ones that make use of group theory. Finally, I will give more details about the EKR theorem for permutation groups, where the role of group theory profound.

You can watch the talk here.

This talk was given on July 20th, 2020