Towards a general theory of Erdős-Ko-Rado combinatorics

2014 Summer.Includes bibliographical references.In 1961, Erdős, Ko, and Rado proved that for a universe of size n ≥ 2k a family of k-subsets whose members pairwise intersect cannot be larger than n-1/k-1. This fundamental result of extremal combinatorics is now known as the EKR theorem for intersecting set families. Since then, there has been a proliferation of similar EKR theorems in extremal combinatorics that characterize families of more sophisticated objects that are largest with respect to a given intersection property. This line of research has given rise to many interesting combinatorial and algebraic techniques, the latter being the focus of this thesis. Algebraic methods for EKR results are attractive since they could potentially give rise to a unified theory of EKR combinatorics, but the state-of-the-art has been shown only to apply to sets, vector spaces, and permutation families. These categories lie on opposite ends of the stability spectrum since the stabilizers of sets and vector spaces are large as possible whereas the stabilizer of a permutation is small as possible. In this thesis, we investigate a category that lies somewhere in between, namely, the perfect matchings of the complete graph. In particular, we show that an algebraic method of Godsil's can be lifted to the more general algebraic framework of Gelfand pairs, giving the first algebraic proof of the EKR theorem for intersecting families of perfect matchings as a consequence. There is strong evidence to suggest that this framework can be used to approach the open problem of characterizing the maximum t-intersecting families of perfect matchings, whose combinatorial proof remains illusive. We conclude with obstacles and open directions for extending this framework to encompass a broader spectrum of categories

On Hamilton cycles in graphs defined by intersecting set systems

In 1970 Lov\'asz conjectured that every connected vertex-transitive graph admits a Hamilton cycle, apart from five exceptional graphs. This conjecture has recently been settled for graphs defined by intersecting set systems, which feature prominently throughout combinatorics. In this expository article, we retrace these developments and give an overview of the many different ingredients in the proofs

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum