Perfect Packings in Quasirandom Hypergraphs II

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 Kr(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.