Logical laws for short existential monadic second order sentences about graphs

Abstract

In this paper, we study existential monadic second order (EMSO) properties of undirected graphs. In 2001, J.-M. Le Bars proved that there exists an EMSO sentence about undirected graphs such that the probability that it is true does not converge (here, the probability distribution is uniform over the set of all graphs on the fixed set of vertices). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the 0-1 law holds (every sentence is either true asymptotically almost surely (a.a.s.), or false a.a.s.). We give an example of EMSO sentence with 1 monadic variable without convergence and an example of EMSO sentence with 3 first order variables without convergence.