Souslin quasi-orders and bi-embeddability of uncountable structures

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 $\kappa$.

160 pages, 5 figures; accepted for publication on the Memoirs of the American Mathematical Society; several changes (including title) following the suggestions of the anonymous referee