Perfect Packings in Quasirandom Hypergraphs II

TL;DR

This paper proves that certain quasirandom hypergraphs contain perfect packings of specific subhypergraphs, extending previous results and establishing tight conditions for such packings.

Contribution

It identifies broad classes of hypergraphs where quasirandomness guarantees perfect packings, providing tight conditions and extending known results.

Findings

Quasirandom hypergraphs admit perfect packings of specified subhypergraphs.

The conditions for perfect packings are shown to be tight and cannot be weakened.

The results extend previous work using hypergraph blowup lemmas.

Abstract

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4 and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the following two properties: * for every graph G with V(G)=V(H), at least p proportion of the triangles in G are also edges of H, * for every vertex x of H, the link graph of x is a quasirandom graph with density at least p. Then H has a perfect $K_r^{(3)}$-packing. Moreover, we show that neither hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's hypergraph blowup lemma, with a slightly stronger hypothesis on H.