Perfect Packings in Quasirandom Hypergraphs II

Source 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 of the 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 Blow-up Lemma, with a slightly stronger hypothesis on H .