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Mat. Sb., 2012, Volume 203, Number 7, Pages 95–128 (Mi msb7698)  

This article is cited in 5 scientific papers (total in 5 papers)

A weak zero-one law for sequences of random distance graphs

M. E. Zhukovskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study zero-one laws for properties of random distance graphs. Properties written in a first-order language are considered. For $p(N)$ such that $pN^{\alpha}\to\infty$ as $N\to\infty$, and $(1-\nobreak p)N^{\alpha}\to\infty$ as $N\to\infty$ for any $\alpha>0$, we succeed in refuting the law. In this connection, we consider a weak zero-one $j$-law. For this law, we obtain results for random distance graphs which are similar to the assertions concerning the classical zero-one law for random graphs.
Bibliography: 18 titles.

Keywords: zero-one laws, first-order language, random graphs, distance graphs, Ehrenfeucht game.

DOI: https://doi.org/10.4213/sm7698

Full text: PDF file (738 kB)
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English version:
Sbornik: Mathematics, 2012, 203:7, 1012–1044

Bibliographic databases:

UDC: 519.179.4
MSC: Primary 05C80; Secondary 03C13, 60F20
Received: 25.02.2010 and 21.08.2011

Citation: M. E. Zhukovskii, “A weak zero-one law for sequences of random distance graphs”, Mat. Sb., 203:7 (2012), 95–128; Sb. Math., 203:7 (2012), 1012–1044

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. N. Popova, “Zero-one laws for random distance graphs with vertices in $\{0,1\}^n$”, Dokl. Math., 90:2 (2014), 535–538  crossref  crossref  zmath  isi  elib  elib  scopus
    2. M. E. Zhukovskii, A. M. Raigorodskii, “Random graphs: models and asymptotic characteristics”, Russian Math. Surveys, 70:1 (2015), 33–81  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. S. N. Popova, “Zero-one law for random subgraphs of some distance graphs with vertices in $\mathbb Z^n$”, Sb. Math., 207:3 (2016), 458–478  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. S. N. Popova, “Zero-one laws for random graphs with vertices in a Boolean cube”, Siberian Adv. Math., 27:1 (2017), 26–75  mathnet  crossref  crossref  mathscinet  elib
    5. A. V. Burkin, M. E. Zhukovskii, “Small subgraphs and their extensions in a random distance graph”, Sb. Math., 209:2 (2018), 163–186  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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