Measurable chromatic numbers
@article{Miller2008MeasurableCN, title={Measurable chromatic numbers}, author={Benjamin D. Miller}, journal={Journal of Symbolic Logic}, year={2008}, volume={73}, pages={1139 - 1157} }
Abstract We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ]ℕ, although its Borel chromatic number is ℵ0. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for…
9 Citations
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References
SHOWING 1-10 OF 18 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…
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…
Ends of graphed equivalence relations, I
- Mathematics
- 2008
Given a countable Borel equivalence relation E on a Polish space, we show: (1) E admits an endless graphing if and only if E is smooth; (2) E admits a locally finite single-ended graphing if and only…
Additivity of measure implies additivity of category
- Mathematics
- 1984
In this paper it is proved that 2'-additivity of category follows from 2'-additivity of measure, and a combinatorial characterization of additivity of measure is found. DEFINITIONS. For abbreviation…
Descriptive Kakutani equivalence
- Mathematics
- 2010
We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly…
Basis theorems for non-Potentially closed Sets and graphs of uncountable Borel chromatic number
- MathematicsJ. Math. Log.
- 2008
We show that there is an antichain basis for neither (1) the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor (2) the class of analytic graphs of…
A Glimm-Effros dichotomy for Borel equivalence relations
- Mathematics
- 1990
A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in…
Means on equivalence relations
- Mathematics
- 2008
Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also…
Countable Borel Equivalence Relations
- MathematicsJ. Math. Log.
- 2002
This paper develops the foundations of the descriptive set theory of countable Borel
equivalence relations on Polish spaces with particular emphasis on the study of hyper-finite, amenable, treeable…
BOREL AUTOMORPHISMS WITH NO FINITE INVARIANT MEASURE
- Mathematics
- 1998
An uncountable family of non-isomorphic Borel automorphisms which do not preserve any finite measure is presented.