Uniformity, universality, and computability theory

Marks, Andrew

doi:10.1142/s0219061317500039

Cited by 12 publications

(15 citation statements)

References 42 publications

(76 reference statements)

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“…Given an E F2 -invariant subset A ⊆ H(F F2 2 ), we define A ∈ U if and only if player II wins the game G A∪C defined in the proof of Lemma 1.3. The proof that this defines an ultrafilter is identical to the proof of [13,Lemma 4.9].…”

Section: An Ultrafilter On R/ementioning

confidence: 71%

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Hyperfiniteness and Borel combinatorics

et al. 2019

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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-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also give examples of hyperfinite bounded degree Borel graphs for which the Borel local lemma fails, in contrast to the recent results of Csóka, Grabowski, Máthé, Pikhurko, and Tyros.Related to the Borel Ruziewicz problem, we show there is a continuous paradoxical action of (Z/2Z) * 3 on a Polish space that admits a finitely additive invariant Borel probability measure, but admits no countably additive invariant Borel probability measure. In the context of studying ultrafilters on the quotient space of equivalence relations under AD, we also construct an ultrafilter U on the quotient of E 0 which has surprising complexity. In particular, Martin's measure is Rudin-Kiesler reducible to U .We end with a problem about whether every hyperfinite bounded degree Borel graph has a witness to its hyperfiniteness which is uniformly bounded below in size.

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Section: An Ultrafilter On R/ementioning

confidence: 71%

“…Here, for example, it is open whether every ultrafilter on 2 N /E 0 is Rudin-Kiesler above U L or U C . (See [13,Section 4] for further discussion and a definition of Rudin-Kiesler reducibility). We show the existence of an ultrafilter on 2 N /E 0 which has surprising complexity:…”

Section: Introductionmentioning

confidence: 99%

Hyperfiniteness and Borel combinatorics

et al. 2019

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“…In particular, this answers a question of Marks [M,end of Section 4.3], who asked for a characterization of when E ∞σ A (σ A a Scott sentence) is smooth. The proof uses ideas from topological dynamics and ergodic theory.…”

Section: Structurability and Model Theorymentioning

confidence: 72%

Structurable equivalence relations

Chen

Kechris

2018

Fund. Math.

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For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K. In particular, we characterize the K such that every K-structurable equivalence relation is smooth, answering a question of Marks. PreliminariesFor general model theory, see [Hod]. For general classical descriptive set theory, see [K95].

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“…Marks's uniformity conjecture (Conjecture 1.4 in [18]). A countable Borel equivalence relation is universal if and only if it is uniformly universal with respect to every way it can be generated.…”

Section: Random Realsmentioning

confidence: 99%

Equivalence of generics

Smythe¹

2018

Preprint

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Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.

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Uniformity, universality, and computability theory

Cited by 12 publications

References 42 publications

Hyperfiniteness and Borel combinatorics

Hyperfiniteness and Borel combinatorics

Structurable equivalence relations

Equivalence of generics

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