ON HOMOMORPHISM GRAPHS

IF 2.8 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi Pub Date : 2024-05-13 DOI:10.1017/fmp.2024.8

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

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引用次数: 0

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.

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上的同态图

我们引入了新类型的有界度非循环 Borel 图实例，并使用 Marks [Mar16] 的确定性方法的广义化，在描述性组合学的背景下研究了它们的组合特性。这种方法适用于分布式计算的 $\mathsf {LOCAL}$ 模型 [BCG+21]。我们的方法统一了该领域以前的成果，同时也产生了新的成果。特别是，为了加强 [TV21] 的主要结果，我们证明了对于 $\Delta>2$ 来说，不可能给出一个简单的无循环 $\Delta $ -regular Borel graphs 的特征，其 Borel 色度数至多为 $\Delta $：这样的图构成了一个 $\mathbf {\Sigma }^1_2$ -complete set。这意味着类似布鲁克斯的定理在 Borel 背景下的强烈失败。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Forum of Mathematics Pi Mathematics-Statistics and Probability

CiteScore

3.50

自引率

0.00%

发文量

审稿时长

19 weeks

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