# Short Monadic Second Order Sentences about Sparse Random Graphs

## Abstract

*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.

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**Submitted**: 11 November 2016

**Accepted**: 18 October 2018

**Published online**: 18 December 2018

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