On the zero–one [formula omitted]-law extensions

Abstract

The presented paper is devoted to the asymptotical behavior of first-order properties of the Erdős–Rényi random graph. In previous works the zero–one k-law was proved. This law describes asymptotical behavior of first-order properties which are expressed by formulae with a quantifier depth bounded by k. The random graph G(N,N−α) obeys the law if α∈(0,1/(k−2)). In this work we find new values of α, which are close to 1, such that G(N,N−α) obeys the zero–one k-law and, therefore, extend the previous result.