Integer and fractional packings in dense 3-uniform hypergraphs

Let J(o) be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph H we define v(jo)(H) to be the maximum size of a set of pairwise triple-disjoint copies of j(o) in H. We say a function psi from the set of copies of j(o) in X to [0, 1] is a fractional J(o)-packing of H if Sigma(gis an element ofe) psi(j) less than or equal to 1 for every triple e of H. Then v(jo)*(H) is defined to be the maximum value of Sigma(gis an element of)((H)(jo)) psi(J) over all fractional J(o)-packings psi of H. We show that v(Jo)*(H) - v(Jo)(H) = o(\V(H)\(3) for all 3-uniform hypergraphs H. This extends the analogous result for graphs, proved by Haxell and Rodl (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 Rbdl (2002)], then almostevery triple is in about the same number of copies of K (3) (the complete 3-uniform hypergraph with k vertices). (C) 2003 Wiley Periodicals, Inc.