First-order properties of bounded quantifier depth of very sparse random graphs

Abstract

We say that a random graph obeys the zero-one -law if every property expressed by a first-order formula with quantifier depth at most holds with probability tending to or . It is known that the random graph obeys the zero-one -law for every and every positive irrational , as well as for all rational which are not of the form (for any positive integer ). It is also known that for all other rational positive , the random graph does not obey the zero-one -law for sufficiently large . In this paper we put and obtain upper and lower bounds for the largest such that the zero-one -law holds.