EMSO(FO$^2$) 0-1 law fails for all dense random graphs

TL;DR

This paper demonstrates that the EMSO(FO$^2$) 0-1 law does not hold for all dense random graphs, showing that certain logical properties do not have a converging probability in $G(n,p)$ for any fixed $p$.

Contribution

It proves the failure of EMSO(FO$^2$) convergence law for binomial random graphs across all constant probabilities, identifying a specific logical sentence with non-converging probability.

Findings

Existential monadic second order sentence with 2 variables has non-converging probability.

The EMSO(FO$^2$) 0-1 law fails for all $p eq 0,1$ in $G(n,p)$.

Convergence law does not hold universally for dense random graphs.

Abstract

In this paper, we disprove EMSO(FO$^2$) convergence law for the binomial random graph $G(n,p)$ for any constant probability $p$. More specifically, we prove that there exists an existential monadic second order sentence with 2 first order variables such that, for every $p\in(0,1)$, the probability that it is true on $G(n,p)$ does not converge.