Souslin quasi-orders and bi-embeddability of uncountable structures

TL;DR

This paper extends known results about the complexity of embeddability relations from countable to uncountable structures, providing new insights into their classification and complexity for various mathematical objects.

Contribution

It generalizes results on quasi-orders and embeddability relations from countable to uncountable structures using $oldsymbol{ ext{kappa}}$-Souslin quasi-orders, broadening the scope of descriptive set theory.

Findings

Complexity results for embeddability of uncountable graphs and lattices

Analysis of isometric embeddability in metric spaces of uncountable density

Classification of Banach spaces based on uncountable density character

Abstract

We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $\kappa = \omega$) for arbitrary $\kappa$-Souslin quasi-orders on any Polish space, for $\kappa$ an infinite cardinal smaller than the cardinality of $\mathbb{R}$. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size $\kappa$, the isometric embeddability relation between complete metric spaces of density character $\kappa$, and the linear isometric embeddability relation between (real or complex) Banach spaces of density $\kappa$.