Integer and fractional packings in dense 3‐uniform hypergraphs

Abstract

AbstractLet 𝒥0 be any fixed 3‐uniform hypergraph. For a 3‐uniform hypergraph ℋ︁ we define ν(ℋ︁) to be the maximum size of a set of pairwise triple‐disjoint copies of 𝒥0 in ℋ︁. We say a function ψ from the set of copies of 𝒥0 in ℋ︁ to [0, 1] is a fractional 𝒥0‐packing of ℋ︁ if ∑𝒥∋e ψ(𝒥) ≤ 1 for every triple e of ℋ︁. Then ν(ℋ︁) is defined to be the maximum value of ∑ ψ(𝒥) over all fractional 𝒥0‐packings ψ of ℋ︁. We show that ν(ℋ︁) − ν(ℋ︁) = o(|V(ℋ︁)| 3) for all 3‐uniform hypergraphs ℋ︁. This extends the analogous result for graphs, proved by Haxell and Rödl (2001), and requires a significant amount of new theory about regularity of 3‐uniform hypergraphs. In particular, we prove a result that we call the Extension Theorem. This states that if a k‐partite 3‐uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and Rödl (2002)], then almost every triple is in about the same number of copies of K (the complete 3‐uniform hypergraph with k vertices). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 248–310, 2003