# 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…

## 10 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.

Of Toasts and Tails

- MathematicsArXiv
- 2021

This work presents an intimate connection among the following fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics: coming from the area of computer science, to view all three perspectives as a part of a common theory of locality.

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…

## References

SHOWING 1-10 OF 27 REFERENCES

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…

A bound on measurable chromatic numbers of locally finite Borel graphs

- Mathematics
- 2016

A graph on a set X is an irreflexive, symmetric set G ⊆ X ×X. Such a graph is locally finite if every point has only finitely many G-neighbors. A (κ-)coloring of such a graph is a function c : X → κ…

Uniformity, universality, and computability theory

- MathematicsJ. Math. Log.
- 2017

A number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations are proved, including the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant borel sets of equivalence Relations that are much finer than Turing equivalence.

The structure of hy-per nite Borel equivalence relations

- Mathematics
- 1994

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not…

Structurable equivalence relations

- Mathematics
- 2016

For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal…

Banach-Tarski paradox using pieces with the property of Baire.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1992

This note solves a problem of Marczewski from 1930 by showing that there is a partition of S2 into sets A1, Ak, B1,..., Bl with a different strong regularity property, the Property of Baire.

Martin’s conjecture and strong ergodicity

- MathematicsArch. Math. Log.
- 2009

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation…

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…

The Banach-Tarski Paradox

- Education
- 1990

Author’s note: This paper was originally written for my Minor Thesis requirement of the Ph.D. program at Harvard University. The object of this requirement is to learn about a body of work that is…