# 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}
}
• Published 7 November 2016
• Mathematics
• Journal of the European Mathematical Society
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 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 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 • Mathematics ArXiv • 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 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 • 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. Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem 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 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. Of Toasts and Tails • Mathematics ArXiv • 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 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. 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