Towards a general theory of Erdős-Ko-Rado combinatorics
dc.contributor.author | Lindzey, Nathan, author | |
dc.contributor.author | Penttila, Tim, advisor | |
dc.contributor.author | Hulpke, Alexander, committee member | |
dc.contributor.author | Boucher, Christina, committee member | |
dc.date.accessioned | 2007-01-03T06:39:38Z | |
dc.date.available | 2007-01-03T06:39:38Z | |
dc.date.issued | 2014 | |
dc.description.abstract | 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. | |
dc.format.medium | born digital | |
dc.format.medium | masters theses | |
dc.identifier | Lindzey_colostate_0053N_12605.pdf | |
dc.identifier.uri | http://hdl.handle.net/10217/83990 | |
dc.language | English | |
dc.language.iso | eng | |
dc.publisher | Colorado State University. Libraries | |
dc.relation.ispartof | 2000-2019 | |
dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
dc.subject | algebraic combinatorics | |
dc.subject | extremal combinatorics | |
dc.subject | Erdős-Ko-Rado theorems | |
dc.subject | association schemes | |
dc.subject | algebraic graph theory | |
dc.title | Towards a general theory of Erdős-Ko-Rado combinatorics | |
dc.type | Text | |
dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
thesis.degree.discipline | Mathematics | |
thesis.degree.grantor | Colorado State University | |
thesis.degree.level | Masters | |
thesis.degree.name | Master of Science (M.S.) |
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