A geometric theory for hypergraph matching

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-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.

Polynomial-time perfect matchings in dense hypergraphs

“…The rest of the paper is devoted to the proof of Theorem 1.10, beginning with Section 3 in which we briefly sketch the ideas of the proof and its key lemmas (Lemmas 7.1 and 7.6). In Section 4 we introduce a number of necessary preliminaries, including results from [17], some convex geometry and well-known probabilistic tools. Section 5 focuses on an important definition, that of 'robust maximality', and some of its properties; this turns out to be the correct notion for the key lemmas.…”

“…In this subsection we describe a central theorem from [17] that plays a key role in this paper. First we need the following definitions.…”

“…There is a large literature on minimum degree conditions for perfect matchings in hypergraphs, see e.g. [1,2,6,7,10,15,17,18,19,20,22,23,24,25,26,28,30,31,32] and the survey by Rödl and Ruciński [27] for details.…”

“…Approximate divisibility barriers. Our starting point will be (a special case of) a result of Keevash and Mycroft [17] on approximate divisibility barriers. First we introduce a less restrictive degree assumption, which follows from the assumption in Theorem 1.1 when γ > 0 is small.…”

“…A result from [17] (stated here as Theorem 4.6), combined with Lemma 7.3, implies that under Setup 1.7, if H does not contain a perfect matching then we can delete o(n k ) edges from H to obtain a subgraph H ′ for which there exists a partition P of V (H ′ ) such that i P (V (H ′ )) / ∈ L P (H ′ ). Thus if H is far from a divisibility barrier then it has a perfect matching.…”