Martin’s conjecture and strong ergodicity

Thomas, Simon

doi:10.1007/s00153-009-0148-0

Cited by 16 publications

(13 citation statements)

References 19 publications

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“…In [Tho09], Thomas shows that Martin's conjecture implies that the Borel complexity of any weakly universal countable Borel equivalence relation must concentrate off of a conull set with respect to any Borel probability measure. In [Mar13b], Marks shows that the Borel complexity of any universal K-structurable countable Borel equivalence relation is achieved on a null set with respect to any Borel probability measure.…”

Section: Proposition 21 Suppose That There Exists a Class-bijectivementioning

confidence: 99%

Borel structurability on the 2-shift of a countable group

Seward

Tucker-Drob

2016

Annals of Pure and Applied Logic

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Section: Proposition 21 Suppose That There Exists a Class-bijectivementioning

confidence: 99%

Borel structurability on the 2-shift of a countable group

Seward

Tucker-Drob

2016

Annals of Pure and Applied Logic

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“…(Here, a subset of 2 ω is said to be a cone if it is

\leq_{T}

‐upwards closed). We shall require only the following consequence of Martin's Conjecture (cf., e.g., [, Theorem 2.1(i)]): Lemma Assuming Martin's conjecture, if

f : \equiv_{T} \to E_{0}

is a Borel homomorphism then there exists a cone C such that

f (C)

is contained in a single E 0 ‐class.…”

Section: A General Framework For Combinatorial Propertiesmentioning

confidence: 99%

“…Using this, Thomas proved in [, Theorem 5.2] that Martin's Conjecture implies that

\equiv_{T}

is not Borel bounded. We now show that his argument applies to any (nontrivial) property corresponding to an invariant relation.…”

Section: A General Framework For Combinatorial Propertiesmentioning

confidence: 99%

Cardinal characteristics and countable Borel equivalence relations

Coskey

Schneider

2017

Mathematical Logic Qtrly

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Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number frakturb. We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding to the splitting number frakturs coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.

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“…Such ultrafilters arise naturally from E-ergodic probability measures on X and E-generically ergodic Polish topologies on X. More recently, work started by Simon Thomas in [20] and continued in [12] and [11] has used ultrafilters related to Martin's ultrafilter on the Turing degrees to derive structural consequences about universal equivalence relations of various kinds. For example, in [12], these ultrafilters from computability were used to show that if E is a universal countable Borel equivalence relation on a standard Borel space X, and A ⊆ X is Borel, then either E ↾ A is universal or E ↾ (X \ A) is universal.…”

Section: Ultrafilters Universality and Relativizationmentioning

confidence: 99%