# Hyperfiniteness and Borel combinatorics

@article{Conley2019HyperfinitenessAB, title={Hyperfiniteness and Borel combinatorics}, author={Clinton T. Conley and Steve Jackson and Andrew S. Marks and Brandon Seward and Robin D. Tucker-Drob}, journal={Journal of the European Mathematical Society}, year={2019} }

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d…

## 11 Citations

### Borel combinatorics of locally finite graphs

- MathematicsBCC
- 2021

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.

### Borel asymptotic dimension and hyperfinite equivalence relations

- Mathematics
- 2020

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In…

### Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics

- Mathematics
- 2020

In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made…

### On Homomorphism Graphs

- MathematicsArXiv
- 2021

It is shown that for ∆ > 2 it is impossible to give a simple characterization of acyclic ∆-regular Borel graphs with Borel chromatic number at most ∆: such graphs form a Σ12-complete set, implying a strong failure of Brooks’-like theorems in the Borel context.

### Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

- MathematicsArXiv
- 2021

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.

### Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

- MathematicsITCS
- 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.

### Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem

- MathematicsIsrael Journal of Mathematics
- 2019

Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ↷ ( X, μ ) and a map f ∈ L 1 ( X, μ ), and to compare the…

### Moser-Tardos Algorithm with small number of random bits

- Mathematics, Computer ScienceArXiv
- 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.

### Measure Asymptotic Separation Index and Hyperfiniteness

- Mathematics
- 2021

In this note, we show that modulo a null set, hyperfiniteness, finite asymptotic separation index (asi), asi 1, and existence of Borel toast are all equivalent. This is of interest as several of the…

### MEASURABLE GRAPH COMBINATORICS

- Mathematics
- 2022

. We survey some recent results in the theory of measurable graph combinatorics. We also discuss applications to the study of hyperfiniteness and measurable equidecompositions.

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