Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201

A Geometric Theory for Hypergraph Matching

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, our main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, we apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical we defer it to a subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical Society. 101 pages. v2: minor changes including some additional diagrams and passages of expository tex

Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems

2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12

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