Degree spectra of analytic complete equivalence relations

TL;DR

This paper explores the complexity of bi-embeddability relations on graphs, showing that elementary bi-embeddability is an analytic complete equivalence relation and analyzing the spectra of these relations.

Contribution

It establishes a Borel reduction from embeddability to elementary embeddability on graphs and characterizes the spectra of bi-embeddability relations.

Findings

Elementary bi-embeddability on graphs is an analytic complete equivalence relation.

Every bi-embeddability spectrum of a graph is a jump spectrum of an elementary bi-embeddability spectrum.

The paper provides a detailed analysis of the degree spectra of these relations.

Abstract

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by this relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a analytic complete equivalence relation. We then investigate the algorithmic properties of this reduction to show that every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph.