Extension of zero-one k-law

TL;DR

This paper investigates the asymptotic probabilities of first-order properties in random graphs G(N,p) with specific edge probability decay, identifying parameter ranges where the zero-one k-law holds.

Contribution

It extends the zero-one k-law to new parameter ranges for the random graph G(N,p) with p defined by lnp=-a ln N, for a>0.

Findings

Identifies parameter ranges where zero-one k-law applies.

Determines critical values of a for the law to hold.

Provides asymptotic probability analysis for first-order properties.

Abstract

We study an asymptotic behavior of the probabilities of first-order properties of random graph G(N,p) in the article. We conider p such that lnp=-alnN, a>0. We find values of parameter a from (1-exp(ln2(1-k)),1) such that the random graph obeys zero-one k-law.