Perfect Packings in Quasirandom Hypergraphs

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.