Baire-class ξ colorings: the first three levels
نویسندگان
چکیده
The G0-dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the G0-dichotomy for Σ 0 ξ-measurable countable colorings when ξ ≤ 3. A Σ 0 ξ-measurable countable coloring gives a covering of the diagonal consisting of countably many Σξ squares. This leads to the study of countable unions of Σξ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when ξ≤2. 2010 Mathematics Subject Classification. Primary: 03E15, Secondary: 54H05
منابع مشابه
Ranks on the Baire Class Ξ Functions
In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class ξ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness ...
متن کاملCharacterization of Order Types of Pointwise Linearly Ordered Families of Baire Class 1 Functions
In the 1970s M. Laczkovich posed the following problem: Let B1(X) denote the set of Baire class 1 functions defined on a Polish space X equipped with the pointwise ordering. Characterize the order types of the linearly ordered subsets of B1(X). The main result of the present paper is a complete solution to this problem. We prove that a linear order is isomorphic to a linearly ordered family of ...
متن کاملTopological Aspects of Hurewicz Tests for the Difference Hierarchy
We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Γ is any of the classes Σξ , Π 0 ξ , Dη(Σ 0 ξ) or Ďη(Σ 0 ξ) we find a representative set PΓ ∈ Γ and a Polish topology τΓ such that for every A ∈ Γ̌ from some assumption on the size of A ∩ PΓ we can deduce that A \ PΓ is of second category in the topology τΓ. This allows us to distinguish the levels of the B...
متن کاملMeasurable 3-colorings of Acyclic Graphs
This is the first of two lectures on measurable chromatic numbers given in June 2010 at the University of Barcelona. Our main result here is that acyclic locally finite analytic graphs on Polish spaces admit Baire measurable 3-colorings.
متن کاملBorel chromatic number of closed graphs
We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number two and Baire class ξ chromatic number א0. 2010 Mathematics Subject Classification. Primary: 03E15, Secondary: 54H05
متن کامل