On random sampling in uniform hypergraphs

A k ‐graph \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} ${\mathcal{G}}^{(k)}$ \end{document} on vertex set [ n ] = {1,…, n } is said to be (ρ,ζ)‐uniform if every S ⊆ [ n ] of size s = | S | > ζ n spans (ρ ± ζ) \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} $\binom{s}{k}$ \end{document} edges. A ‘grabbing lemma’ of Mubayi and Rödl shows that this property is typically inherited locally: if \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} ${\mathcal{G}}^{(k)}$ \end{document} is (ρ,ζ)‐uniform, then all but exp{− s 1/ k /20} \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} $\binom{n}{s}$ \end{document} sets \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} $ S \in \binom{[n]}{s}$ \end{document} span (ρ,ζ′)‐uniform subhypergraphs \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs,amssymb}\pagestyle{empty}\begin{document} ${\mathcal{G}}^{(k)}\lbrack S\rbrack$ \end{document} , where ζ′→ 0 as ζ → 0, s ≥ s 0 (ζ′) and n is sufficiently large. In this article, we establish a grabbing lemma for a different concept of hypergraph uniformity, and infer the result above as a corollary. In particular, we improve, in the context above, the error exp{− s 1/ k /20} to exp{− cs }, for a constant c = c ( k ,ζ′) > 0. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 422–440, 2011