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A geometric theory for hypergraph matching


About this Title

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
Keywords:hypergraphs

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Results and examples
  • Chapter 3. Geometric Motifs
  • Chapter 4. Transferrals
  • Chapter 5. Transferrals via the minimum degree sequence
  • Chapter 6. Hypergraph Regularity Theory
  • Chapter 7. Matchings in $k$-systems
  • Chapter 8. Packing Tetrahedra
  • Chapter 9. The general theory

Abstract


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 -graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi 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.

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