Integer and fractional packings in dense 3-uniform hypergraphs

Let 0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph we define 0( ) to be the maximum size of a set of pairwise triple-disjoint copies of 0 in . We say a function from the set of copies of 0 in to [0, 1] is a fractional 0-packing of if ¥ e ( ) 1 for every triple e of . Then * 0( ) is defined to be the maximum value of ¥ 0 over all fractional 0-packings of . We show that * 0( ) 0( ) o( V( ) ) for all 3-uniform hypergraphs . 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 Kk (3) (the complete 3-uniform hypergraph with k vertices). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 248–310, 2003