Pseudorandom hypergraph matchings

Abstract

A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.

A rainbow blow-up lemma for almost optimally bounded edge-colourings

“…For the embedding not to get stuck, we need to find in 2 not just any , but a good one. To achieve this, we make use of a general hypergraph matching theorem (Theorem 4.3) proved recently by the authors in [13], which guarantees a matching M in H that is in many ways 'random-like'. This will allow us to find an almost-perfect rainbow matching for which the updated candidacy graph 3 will have the desired properties.…”

“…As sketched in Section 3, we use hypergraph matchings to model rainbow embeddings. In this section, we introduce a theorem from [13] on 'pseudorandom' hypergraph matchings (Theorem 4.3), which will play an important role in Section 6.…”

“…Following the seminal result of Rödl [41] on approximate Steiner systems, Pippenger observed that any almost regular uniform hypergraph with small codegrees has an almost perfect matching. In [13], the authors proved a tool that allows to obtain 'pseudorandom' matchings in this setting. To make this more precise, we define for a hypergraph H and vertices , ∈ (H) the degree deg Suppose for simplicity that we are given a -regular hypergraph and want to find an (almost) perfect matching M. Moreover, we wish M to be 'pseudorandom': that is, to have certain properties that we expect from an idealized random matching.…”

“…The following theorem asserts that a hypergraph with small codegrees has a matching that is pseudorandom in this sense. We refer the interested reader to [13] for more information on the preceding results and further variants and applications of Theorem 4.3. In particular, Ehard and Joos recently used a more general version of Theorem 4.3 to give a short proof of the blow-up lemma for approximate decompositions [14].…”

Decomposing hypergraphs into cycle factors

“…For large L (but which does not grow with n) the k-graph H 1 has a fractional decomposition into cycles of length L, by a recent result in [ 15 ] (see Theorem 3.4 ). Next, we exploit a result about hypergraph matchings with pseudorandom properties [ 5 ] (see Theorem 3.5 and Corollary 3.6 ) to turn this fractional decomposition of H 1 into edge-disjoint collections P 1 , . .…”