• Corpus ID: 243831677

On Homomorphism Graphs

@article{Brandt2021OnHG,
title={On Homomorphism Graphs},
author={Sebastian Brandt and Yi-Jun Chang and Jan Greb{\'i}k and Christoph Grunau and V{\'a}clav Rozhoň and Zolt'an Vidny'anszky},
journal={ArXiv},
year={2021},
volume={abs/2111.03683}
}
• Published 5 November 2021
• Mathematics
• ArXiv
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 [Mar16]. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing [BCG21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for ∆ > 2 it is impossible…
2 Citations

Figures from this paper

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics
• Mathematics
ITCS
• 2022
This approach that borrows techniques from the fields (a), (b) and (c) implies a number of results about possible complexities of finitary factor solutions and helps to view all three perspectives as a part of a common theory of locality.
Moser-Tardos Algorithm with small number of random bits
• Mathematics, Computer Science
ArXiv
• 2022
A deterministic algorithm for finding a satisfying assignment, which in any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables.

References

SHOWING 1-10 OF 77 REFERENCES
Borel combinatorics of locally finite graphs
Some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc are presented.
Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms
A version of the Lovász Local Lemma that can be used to prove the existence of continuous colorings is developed and a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing is established.
BOREL CHROMATIC NUMBERS
• Mathematics
• 1999
We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable
Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms
The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory by introducing the qualitative analogues of the Benjamini-Schramm and local-global graph limit theories for sparse graphs.
Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics
• Mathematics
ITCS
• 2022
This approach that borrows techniques from the fields (a), (b) and (c) implies a number of results about possible complexities of finitary factor solutions and helps to view all three perspectives as a part of a common theory of locality.
Borel version of the Local Lemma
• Mathematics
• 2016
We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying
Minimal definable graphs of definable chromatic number at least three
• Mathematics
Forum of Mathematics, Sigma
• 2021
Abstract We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel
BROOKS’ THEOREM FOR MEASURABLE COLORINGS
• Mathematics
Forum of Mathematics, Sigma
• 2016
We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$ -cliques, then $G$ admits a ${\it\mu}$
The independence ratio of regular graphs
A set I of vertices of a graph G = (V, E) is said to be independent if no two vertices of I are joined by an edge of G. The maximal number of vertices in an independent set is the (vertex)