Logical laws for short existential monadic second order sentences about graphs

TL;DR

This paper investigates the probabilistic behavior of existential monadic second order (EMSO) properties in undirected graphs, providing new examples of EMSO sentences with non-converging probabilities and exploring the 0-1 law conjecture.

Contribution

It presents new EMSO sentences with non-converging probabilities, including one with a single monadic variable and another with three first-order variables, advancing understanding of logical laws in graph properties.

Findings

Existence of EMSO sentence with 1 monadic variable without convergence

Existence of EMSO sentence with 3 first-order variables without convergence

Supports the conjecture on 0-1 law for EMSO sentences with 2 variables

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.