Short Monadic Second Order Sentences about Sparse Random Graphs

In this paper, we study zero-one laws for the Erdös--Rényi 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 $\varphi\in\mathcal{K}$ is either asymptotically almost surely (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 quantifier depth $k$, that is, the length of the longest chain of nested quantifiers in the formula expressing the property. We call zero-one laws for properties of quantifier depth $k$ the zero-one $k$-laws. The main results of this paper concern the zero-one $k$-laws for monadic second order (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.