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 Mat. Zametki, 2015, Volume 97, Issue 2, Pages 203–216 (Mi mz10511)

On the Zero-One 4-Law for the Erdős–Rényi Random Graphs

M. E. Zhukovskii

Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

Abstract: The limit probabilities of the first-order properties of a random graph in the Erdős–Rényi model $G(n,n^{-\alpha})$, $\alpha\in(0,1]$, are studied. Earlier, the author obtained zero-one $k$-laws for any positive integer $k\ge 3$, which describe the behavior of the probabilities of the first-order properties expressed by formulas of quantifier depth bounded by $k$ for $\alpha$ in the interval $(0,1/(k-2)]$ and $k\ge 4$ in the interval $(1-1/2^{k-1},1)$. This result is improved for $k=4$. Moreover, it is proved that, for any $k\ge 4$, the zero-one $k$-law does not hold at the lower boundary of the interval $(1-1/2^{k-1},1)$.

Keywords: zero-one $4$-law, zero-one $k$-law, Erdős–Rényi random graph, first-order property.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-0061212-01-00683-à Ministry of Education and Science of the Russian Federation ÌÄ-6277.2013.1ÌÊ-2184.2014.1 This work was supported by the Russian Foundation for Basic Research (grants nos. 13-01-00612 and 12-01-00683-a) and by the programs for support of young candidates and doctors of science (grants nos. MD-6277.2013.1 and MC-2184.2014.1).

DOI: https://doi.org/10.4213/mzm10511

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English version:
Mathematical Notes, 2015, 97:2, 190–200

Bibliographic databases:

UDC: 519.179.4
Revised: 18.09.2014

Citation: M. E. Zhukovskii, “On the Zero-One 4-Law for the Erdős–Rényi Random Graphs”, Mat. Zametki, 97:2 (2015), 203–216; Math. Notes, 97:2 (2015), 190–200

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10511
• https://doi.org/10.4213/mzm10511
• http://mi.mathnet.ru/eng/mz/v97/i2/p203

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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. M. E. Zhukovskii, “The spectra of first-order formulae having low quantifier rank”, Russian Math. Surveys, 70:6 (2015), 1176–1178
2. M. E. Zhukovskii, A. D. Matushkin, “Universal Zero-One $k$-Law”, Math. Notes, 99:4 (2016), 511–523
3. Zhukovskii M.E., Ostrovskii L.B., “First-order and monadic properties of highly sparse random graphs”, Dokl. Math., 94:2 (2016), 555–557
4. Spencer J.H., Zhukovskii M.E., “Bounded quantifier depth spectra for random graphs”, Discrete Math., 339:6 (2016), 1651–1664
5. M. E. Zhukovskii, L. B. Ostrovskii, “First-order properties of bounded quantifier depth of very sparse random graphs”, Izv. Math., 81:6 (2017), 1155–1167
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