Logical laws for short existential monadic second-order sentences about graphs

Abstract

In 2001, Le Bars proved that there exists an existential monadic second-order (EMSO) sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence (even for [Formula: see text]).