Upper bound for the minimal quantifier depth of the first part of a monadic second-order sentence without asymptotic probability

TL;DR

This paper establishes an upper bound on the minimal quantifier depth of a specific monadic second-order sentence related to the extension grid axiom in random graph models, advancing understanding of logical complexity in probabilistic structures.

Contribution

It provides a new upper bound for the quantifier depth of a monadic second-order sentence expressing the extension grid axiom in Erdős-Rényi random graphs.

Findings

Upper bound for quantifier depth established

Applies to monadic second-order sentences without asymptotic probability

Focuses on the extension grid axiom in random graphs

Abstract

In this paper we found an upper bound for the minimal quantifier depth of the first part of a monadic second-order sentence without asymptotic probability described by Jerzy Tyszkiewicz, which express the extension grid axiom in the Erd\H{o}s-R\'enyi model of random graphs $G(n,n^{-\alpha})$ for some irrational $\alpha$.