Perfect packings in quasirandom hypergraphs I

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 -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.

Hamilton cycles in quasirandom hypergraphs

“…Proof of Proposition By Proposition it suffices to show that for any μ > 0 there exists n 0 such that if then the random graph satisfies properties (a) and (b) of Proposition with positive probability. For the case k = 3 short proofs of these statements were given in , Lemmas 20 and 21], and similar arguments hold for any (we omit the details).…”

“…Since a clique and an independent set in an ℓ ‐graph can have at most ℓ –1 vertices in common, has the desired property that the intersection of any edge containing x and any edge not containing x has size at most ℓ –1. Moreover, standard probabilistic arguments similar to those of , Lemmas 20 and 21] show that, for any fixed μ > 0, with high probability is indeed an k ‐graph with for any .…”

“…This section contains proofs of the three key lemmas: the connecting lemma, the absorbing path lemma, and the path cover lemma. For both the connecting lemma and absorbing path lemma, the key element is an extension lemma for quasirandom hypergraphs proved by Lenz and Mubayi . We actually only need the following special case of the lemma, which is obtained from the full version , Lemma ] by taking ; it is easily checked that Eqs.…”

“…In this section we prove Proposition using the following construction, which was presented for 3‐graphs in . Construction For integers ℓ, k and n with , we form a random k ‐graph on n vertices as follows.…”

“…assuming also that H satisfies the minimum ℓ ‐degree condition ). Indeed, by adapting a construction of Lenz and Mubayi we prove the following proposition. Proposition Let k and ℓ be integers such that and k– ℓ divides k. Then for any μ > 0 there exists n 0 such that for any which is divisible by 2k, there is a k‐graph H on n vertices such that a. H is ‐dense , b .…”