A geometric theory for hypergraph matching
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Colleges, School and Institutes
- Mathematical Institute, University of Oxford
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.
A geometric theory for hypergraph matching by Peter Keevash and Richard Mycroft Publication: Memoirs of the American Mathematical Society Publication Year 2015: Volume 233, Number 1098 ISBNs: 978-1-4704-0965-4 (print); 978-1-4704-1966-0 (online) DOI: http://dx.doi.org/10.1090/memo/1098 Published electronically: May 19, 2014
|Title of host publication||A geometric theory for hypergraph matching|
|Publication status||Published - 2015|