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First-order and monadic properties of highly sparse random graphs

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Abstract

A random graph is said to obey the (monadic) zero–one k-law if, for any property expressed by a first-order formula (a second-order monadic formula) with a quantifier depth of at most k, the probability of the graph having this property tends to either zero or one. It is well known that the random graph G(n, n –α) obeys the (monadic) zero–one k-law for any k ∈ ℕ and any rational α > 1 other than 1 + 1/m (for any positive integer m). It is also well known that the random graph does not obey both k-laws for the other rational positive α and sufficiently large k. In this paper, we obtain lower and upper bounds on the largest at which both zero–one k-laws hold for α = 1 + 1/m.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700, Russia

    M. E. Zhukovskii & L. B. Ostrovskii

  2. RUDN University, Moscow, 117198, Russia

    M. E. Zhukovskii

Authors
  1. M. E. Zhukovskii
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  2. L. B. Ostrovskii
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Corresponding author

Correspondence to M. E. Zhukovskii.

Additional information

Original Russian Text © M.E. Zhukovskii, L.B. Ostrovskii, 2016, published in Doklady Akademii Nauk, 2016, Vol. 470, No. 5, pp. 499–501.

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Zhukovskii, M.E., Ostrovskii, L.B. First-order and monadic properties of highly sparse random graphs. Dokl. Math. 94, 555–557 (2016). https://doi.org/10.1134/S1064562416050240

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  • DOI: https://doi.org/10.1134/S1064562416050240

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