Existential monadic second order logic of undirected graphs: a disproof of the Le Bars conjecture

Abstract

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) about undirected graphs. He proved that there exists an EMSO sentence $\varphi$ such that $\mathsf{P}(G_n\models \varphi )$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.