ON HOMOMORPHISM GRAPHS-Reference-Cited by-同舟云学术

ON HOMOMORPHISM GRAPHS

Published:2024 Issue: Volume:12 Page:
ISSN:2050-5086
Container-title:Forum of Mathematics, Pi
language:en
Short-container-title:Forum of Mathematics, Pi

Author:

Brandt Sebastian^ORCID,Chang Yi-Jun^ORCID,Grebík Jan^ORCID,Grunau Christoph^ORCID,Rozhoň Václav^ORCID,Vidnyánszky Zoltán^ORCID

Abstract

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.

Publisher

Cambridge University Press (CUP)

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