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… 
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References

SHOWING 1-10 OF 27 REFERENCES
BOREL CHROMATIC NUMBERS
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
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
TLDR
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
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
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.
  • R. DoughertyM. Foreman
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1992
TLDR
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
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
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
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
...
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