Year: 2024
Type: article
Abstract: We introduce new types 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 [Mar16]. The motivation for the construction comes from the adaptation of this method to the $\mathsf {LOCAL}$ model of distributed computing [BCG+21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, strengthening the main result of [TV21], 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.
Source: Forum of Mathematics Pi
DOI: 10.1017/fmp.2024.8
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,590
Funders: California Institute of Technology, Leverhulme Trust, University of Warwick, Austrian Science Fund, Eidgenössische Technische Hochschule Zürich, Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, ETH Zürich Foundation, National Research, Development and Innovation Office, H2020 European Research Council
OpenAlex ID: W3212905165