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 language | English (US) |
|---|---|
| Pages (from-to) | 422-440 |
| Number of pages | 19 |
| Journal | Random Structures and Algorithms |
| Volume | 38 |
| Issue number | 4 |
| DOIs | |
| State | Published - Jul 2011 |
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Keywords
- Hypergraph regularity
- Random sampling
ASJC Scopus subject areas
- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics
Cite this
On random sampling in uniform hypergraphs. / Czygrinow, Andrzej; Nagle, Brendan.
In: Random Structures and Algorithms, Vol. 38, No. 4, 07.2011, p. 422-440.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On random sampling in uniform hypergraphs
AU - Czygrinow, Andrzej
AU - Nagle, Brendan
PY - 2011/7
Y1 - 2011/7
N2 - 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.
AB - 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.
KW - Hypergraph regularity
KW - Random sampling
UR - http://www.scopus.com/inward/record.url?scp=79956078752&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79956078752&partnerID=8YFLogxK
U2 - 10.1002/rsa.20326
DO - 10.1002/rsa.20326
M3 - Article
AN - SCOPUS:79956078752
VL - 38
SP - 422
EP - 440
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
SN - 1042-9832
IS - 4
ER -