A Geometric Theory for Hypergraph Matching

TL;DR

This paper introduces a geometric framework for hypergraph matchings, identifying barriers to perfect matchings and establishing minimum degree conditions for near-perfect matchings, with applications to open problems in hypergraph packings.

Contribution

It develops a geometric theory for hypergraph matchings, introduces simplicial complexes with degree sequences, and applies the theory to solve open problems in hypergraph packings.

Findings

Determines the best possible minimum degree sequence for almost perfect matchings.

Establishes a stability property linking the absence of perfect matchings to geometric barriers.

Proves exact thresholds for packing tetrahedra and asymptotic results for Fischer's conjecture.

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 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.