Short monadic second order sentences about sparse random graphs

TL;DR

This paper investigates zero-one laws for monadic second order properties of sparse Erdős–Rényi graphs, identifying specific conditions under which these laws hold or fail for different quantifier depths.

Contribution

It determines all values of lpha>0 where the zero-one 3-law for MSO properties fails and shows infinitely many such lpha for the 4-law, advancing understanding of logical properties in sparse graphs.

Findings

Zero-one 3-law for MSO properties fails at specific lpha values.

Infinitely many lpha values where the zero-one 4-law does not hold.

Analysis of graph property evolution relevant to logical law failures.

Abstract

In this paper, we study zero-one laws for the Erd\H{o}s--R\'{e}nyi random graph model $G(n,p)$ in the case when $p = n^{-\alpha}$ for $\alpha>0$. For a given class $\mathcal{K}$ of logical sentences about graphs and a given function $p=p(n)$, we say that $G(n,p)$ obeys the zero-one law (w.r.t. the class $\mathcal{K}$) if each sentence $\varphi\in\mathcal{K}$ either a.a.s. true or a.a.s. false for $G(n,p)$. In this paper, we consider first order properties and monadic second order properties of bounded \textit{quantifier depth} $k$, that is, the length of the longest chain of nested quantifiers in the formula expressing the property. Zero-one laws for properties of quantifier depth $k$ we call the \textit{zero-one $k$-laws}. The main results of this paper concern the zero-one $k$-laws for monadic second order properties (MSO properties). We determine all values $\alpha>0$, for which the zero-one $3$-law for MSO properties does not hold. We also show that, in contrast to the case of the $3$-law, there are infinitely many values of $\alpha$ for which the zero-one $4$-law for MSO properties does not hold. To this end, we analyze the evolution of certain properties of $G(n,p)$ that may be of independent interest.