Long Strong Chains of Subsets of $\omega_1$

TL;DR

This paper demonstrates the forcing of a chain of length ω₃ in the set of uncountable subsets of ω₁, using symmetric systems of models, advancing previous results in the field.

Contribution

It introduces a new forcing construction involving symmetric systems of models to produce longer chains in [ω₁]^{ω₁}, improving prior work by Koszmider and Veličković-Venturi.

Findings

Established the existence of a chain of length ω₃ in [ω₁]^{ω₁}

Used symmetric systems of models as side conditions in forcing

Improved previous bounds on such chains

Abstract

We force the existence of a chain of length $\omega_3$ in $[\omega_1]^{\omega_1}$ increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veli\v{c}kovi\'{c}-Venturi.