On the zero-one 4-law for the Erdős-Rényi random graphs

Abstract

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