Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below.
Copy To Clipboard
Successfully Copied!
A Geometric Theory for Hypergraph Matching
 
Peter Keevash Queen Mary University of London, United Kingdom
Richard Mycroft University of Birmingham, United Kingdom
Front Cover for A Geometric Theory for Hypergraph Matching
Available Formats:
Electronic ISBN: 978-1-4704-1966-0
Product Code: MEMO/233/1098.E
95 pp 
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Front Cover for A Geometric Theory for Hypergraph Matching
Click above image for expanded view
  • Front Cover for A Geometric Theory for Hypergraph Matching
  • Back Cover for A Geometric Theory for Hypergraph Matching
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
95 pp 
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2332014
    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.

  • Table of Contents
     
     
    • 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
  • Request Review Copy
  • Get Permissions
Volume: 2332014
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
Please select which format for which you are requesting permissions.