Complexity of Finite Borel Asymptotic Dimension

TL;DR

This paper investigates the complexity of the class of locally finite Borel graphs with finite Borel asymptotic dimension, establishing its high descriptive set-theoretic complexity and analyzing related homomorphism problems.

Contribution

It provides a combinatorial characterization of finite Borel asymptotic dimension and classifies the complexity of associated digraph homomorphism problems.

Findings

The set of such graphs is $oldsymbol{ ext{ extbf{ extit{ extSigma}}}}^1_2$-complete.

A combinatorial characterization for graphs generated by a single Borel function.

Classified the complexities of digraph homomorphism problems for this class.

Abstract

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.