Monadic second-order properties of very sparse random graphs

TL;DR

This paper investigates the probabilities of certain logical properties in very sparse Erdős–Rényi random graphs, establishing new zero-one laws for both first order and monadic second order logics when the edge probability exponent exceeds 1.

Contribution

It extends zero-one law results to monadic second order properties in sparse random graphs for the case when a > 1, which was previously unexplored.

Findings

Established new zero-one laws for FO and MSO properties when a > 1.

Analyzed monadic second order equivalence classes in sparse graphs.

Extended understanding of logical property probabilities in very sparse regimes.

Abstract

We study asymptotical probabilities of first order and monadic second order properties of Erdos-Renyi random graph G(n,n^{-a}). The random graph obeys FO (MSO) zero-one k-law if for any first order (monadic second order) formulae it is true for G(n,n^{-a}) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a < 1. We obtain new zero-one k-laws (both for first order and monadic second order languages) when a > 1. Proofs of these results are based on the existed study of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.