Universal Zero-One $k$--Law

TL;DR

This paper investigates the probabilities of first-order properties in random graphs, establishing explicit intervals where the Zero-One $k$-Law applies, especially near rational points, and analyzing the asymptotic behavior of these intervals.

Contribution

It provides explicit intervals for the Zero-One $k$-Law near rational points and characterizes their asymptotic size relative to rational numbers.

Findings

Identified explicit intervals where Zero-One $k$-Law holds near rational points.

Proved asymptotic behavior of interval lengths related to rational numbers with numerator ≤ 2.

Established the relationship between interval size and rational points for large n.

Abstract

In this paper the limit probabilities of first-order properties are studied. The random graph $G(n,p)$ {\it obeys Zero-One $k$-Law} if for each first-order property with quantifier depth not greater than $k$ its probability tends to 0 or tends to 1. We found an explicit interval to the left of any rational point on which the Zero-One $k$-Law holds. We also proved, that if $t/s$ is a rational number with numerator not greater than 2, then logarithm of our interval's length has the same asymptotics up to a constant factor (when $n\rightarrow\infty$) as logarithm of the biggest interval with right end at $(t/s)$ on which Zero-One $k$-Law holds.