Existential monadic second order convergence law fails on sparse random graphs

TL;DR

This paper demonstrates that the existential monadic second order convergence law does not hold for certain sparse random graphs, specifically for the binomial model with parameters depending on n and alpha.

Contribution

It proves the failure of the existential monadic second order convergence law in a new class of sparse random graphs, expanding understanding of logical properties in probabilistic graph models.

Findings

Existential monadic second order convergence law fails for G(n, n^(-α))

Failure occurs for all α in (0,1)

Results extend the understanding of logical laws in sparse graphs

Abstract

In the paper, we prove that existential monadic second order convergence law fails for the binomial random graph $G(n,n^{-\alpha})$ for every $\alpha\in(0,1)$.