Integer and fractional packings in dense 3-uniform hypergraphs

  • Authors:

  • Penny Haxell;Brendan Nagle;Vojtěch Rödl

  • Affiliations:

  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1;School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia;Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2003

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Abstract

Let I0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph H we define vI0 (H) to be the maximum size of a set of pairwise triple-disjoint copies of I0 in H. We say a function ψ from the set of copies of I0 in H to [0, 1] is a fractional I0-packing of H if ΣI ∋ e ψ(I) ≤ 1 for every triple e of H. Then v* I 0 (H) is defined to be the maximum value of Σ I ∈ (H I o) ψ(I) over all fractionalI0-packings ψ of H. We show that vI0* (H) - v I0 (H) = o(|V(H)| 3) for all 3-uniform hypergraphs H. 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).