On random sampling in uniform hypergraphs

Andrzej Czygrinow, Brendan Nagle

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

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| > ζn 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{-s1/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 ≥ s0(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{-s1/k/20} to exp{-cs}, for a constant c = c(k,ζ') > 0.

Original languageEnglish (US)
Pages (from-to)422-440
Number of pages19
JournalRandom Structures and Algorithms
Volume38
Issue number4
DOIs
StatePublished - Jul 1 2011

Keywords

  • Hypergraph regularity
  • Random sampling

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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