A geometric theory for hypergraph matching
Author(s)
Bibliographic Information
A geometric theory for hypergraph matching
（Memoirs of the American Mathematical Society, no. 1098）
American Mathematical Society, c2014
Available at 8 libraries
Note
"January 2015, volume 233, number 1098 (fourth of 6 numbers)."
Includes bibliographical references (p. 93-95) and index
Description and Table of Contents
Description
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-Szemeredi 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
Introduction
Results and examples
Geometric motifs
Transferrals
Transferrals via the minimum degree sequence
Hypergraph regularity theory
Matchings in k-systems
Packing tetrahedra
The general theory
Bibliography
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