Abstract
The limit probabilities of the first-order properties of a random graph in the Erdős–Rényi model G(n, n −α), α ∈ (0, 1), are studied. A random graph G(n, n −α) is said to obey the zero-one k-law if, given any property expressed by a formula of quantifier depth at most k, the probability of this property tends to either 0 or 1. As is known, for α = 1− 1/(2k−1 + a/b), where a > 2k−1, the zero-one k-law holds. Moreover, this law does not hold for b = 1 and a ≤ 2k−1 − 2. It is proved that the k-law also fails for b > 1 and a ≤ 2k−1 − (b + 1)2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Janson, T. Luczak, and A. Rucinski, Random Graphs, in Wiley-Intersci. Ser. Discrete Math. Optim. (New York, Wiley, 2000).
M. E. Zhukovskii and A. M. Raigorodskii, “Random graphs: Models and asymptotic characteristics,” UspekhiMat. Nauk 70 1, 35–88 (2015) [RussianMath. Surveys 70 1, 33–81 (2015)].
N. K. Vereshchagin and A. Shen’, Languages and Calculi (MTsNMO, Moscow, 2000) [in Russian].
M. E. Zhukovskii, “Zero-One laws for first-order formulas with bounded quantifier depth,” Dokl. Ross. Akad. Nauk 436 1, 14–18 (2011) [Dokl. Math. 83 1, 8–11 (2011)].
M. Zhukovskii, “Zero-one k-law,” Discrete Math. 312 10, 1670–1688 (2012).
M. E. Zhukovskii, “Extension of the zero-one k-law,” Dokl. Ross. Akad. Nauk 454 1, 23–26 (2014) [Dokl. Math. 89 1, 16–19 (2014)].
M. E. Zhukovskii, “The largest critical point in the zero-one k-law,” Mat. Sb. 206 4, 13–34 (2015) [Sb. Math. 206 4, 489–509 (2015)].
S. Shelah and J. Spencer, “Zero-one laws for sparse randomgraphs,” J. Amer. Math. Soc. 1, 97–115 (1988).
Yu. V. Glebskii, D. I. Kogan, M. I. Liogon’kii, and V. A. Talanov, “Volume and fraction of satisfiability of formulas of the lower predicate calculus,” Kibernetika, No. 2, 17–27 (1969).
R. Fagin, “Probabilities in finite models,” J. Symbolic Logic 41 1, 50–58 (1976).
B. Bollobás, “Threshold functions for small subgraphs,” Math. Proc. Cambridge Philos. Soc. 90 2, 197–206 (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M. E. Zhukovskii, A. E. Medvedeva, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 3, pp. 342–349.
Rights and permissions
About this article
Cite this article
Zhukovskii, M.E., Medvedeva, A.E. When does the zero-one k-law fail?. Math Notes 99, 362–367 (2016). https://doi.org/10.1134/S0001434616030032
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616030032