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A Geometric Theory for Hypergraph Matching

Peter Keevash Queen Mary University of London, United Kingdom
Richard Mycroft University of Birmingham, United Kingdom
Available Formats:
Electronic ISBN: 978-1-4704-1966-0
Product Code: MEMO/233/1098.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view A Geometric Theory for Hypergraph Matching Peter Keevash Queen Mary University of London, United Kingdom Richard Mycroft University of Birmingham, United Kingdom Available Formats:  Electronic ISBN: 978-1-4704-1966-0 Product Code: MEMO/233/1098.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2332014; 95 pp
MSC: Primary 05;

The authors 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, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching.

Furthermore, their 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, the authors 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-Szemerédi Theorem.

Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.

• Chapters
• 1. Introduction
• 2. Results and examples
• 3. Geometric Motifs
• 4. Transferrals
• 5. Transferrals via the minimum degree sequence
• 6. Hypergraph Regularity Theory
• 7. Matchings in $k$-systems
• 8. Packing Tetrahedra
• 9. The general theory
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Volume: 2332014; 95 pp
MSC: Primary 05;

The authors 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, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching.

Furthermore, their 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, the authors 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-Szemerédi Theorem.

Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.

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