Abstract
The limit probabilities of first-order properties of a random graph in the Erdős–Rényi model G(n, n −α), α ∈ (0, 1), are studied. For any positive integer k ≥ 4 and any rational number t/s ∈ (0, 1), an interval with right endpoint t/s is found in which the zero-one k-law holds (the zero-one k-law describes the behavior of the probabilities of first-order properties expressed by formulas of quantifier depth at most k).Moreover, it is proved that, for rational numbers t/s with numerator not exceeding 2, the logarithm of the length of this interval is of the same order of smallness (as n→∞) as that of the length of the maximal interval with right endpoint t/s in which the zero-one k-law holds.
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Original Russian Text © M. E. Zhukovskii, A. D. Matushkin, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 4, pp. 511–525.
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Zhukovskii, M.E., Matushkin, A.D. Universal zero-one k-law. Math Notes 99, 511–523 (2016). https://doi.org/10.1134/S000143461603024X
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DOI: https://doi.org/10.1134/S000143461603024X