# First order sentences about random graphs: small number of alternations

## TL;DR

This paper investigates the spectra of first order sentences over random graphs, establishing that minimal infinite spectra require at least three quantifier alternations and analyzing the limit points of spectra with quantifier depth four.

## Contribution

It proves that the minimal number of quantifier alternations for sentences with infinite spectra is three and characterizes the limit points of spectra for sentences with quantifier depth four.

## Key findings

- Minimal infinite spectra require at least three quantifier alternations.
- Spectra of sentences with quantifier depth 4 have limited limit points.
- Possible limit points for spectra include 1/2 and 3/5.

## Abstract

Spectrum of a first order sentence is the set of all $\alpha$ such that $G(n, n^{-\alpha})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.