Perfect Packings in Quasirandom Hypergraphs

TL;DR

This paper establishes conditions under which quasirandom hypergraphs contain perfect packings of a given hypergraph, extending known results to nonlinear cases and sparse regimes.

Contribution

It proves new perfect packing results for quasirandom hypergraphs, including nonlinear and sparse cases, generalizing previous linear hypergraph results.

Findings

Quasirandom hypergraphs with constant density admit perfect F-packings.

Counterexamples show limitations for certain nonlinear F.

Sparse quasirandom hypergraphs can have perfect matchings based on eigenvalue gaps.

Abstract

Let k >= 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree $\Omega(n^{k-1})$ admits a perfect F-packing. The case k = 2 follows immediately from the blowup lemma of Koml\'os, S\'ark\"ozy, and Szemer\'edi. We also prove positive results for some nonlinear F but at the same time give counterexamples for rather simple F that are close to being linear. Finally, we address the case when the density tends to zero, and prove (in analogy with the graph case) that sparse quasirandom 3-uniform hypergraphs admit a perfect matching as long as their second largest eigenvalue is sufficiently smaller than the largest eigenvalue.