On Homomorphism Graphs
Abstract
We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $\Delta>2$ it is impossible to give a simple characterization of acyclic $\Delta$regular Borel graphs with Borel chromatic number at most $\Delta$: such graphs form a $\mathbf{\Sigma}^1_2$complete set. This implies a strong failure of Brooks'like theorems in the Borel context.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.03683
 Bibcode:
 2021arXiv211103683B
 Keywords:

 Mathematics  Logic;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Mathematics  Combinatorics