A trichotomy theorem in natural models of 𝐴𝐷⁺

Caicedo, Andrés Eduardo; Ketchersid, Richard

doi:10.1090/conm/533/10510

Cited by 14 publications

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“…Their structure was studied by Woodin in [42]. There is the smallest non-linearly-orderable equivalence cardinal, which happens to be E 0 [4]. Beyond that, there is the rich realm of Borel equivalences and a whole zoo of other things that no one has ever seen.…”

Section: Completely Regular Modelsmentioning

confidence: 99%

“…Clearly, if |A| ≤ |B| and the set B is linearly orderable then so is A, and so the concept of linear orderability may serve as a tool for disproving inequalities between sizes of various sets. Under hypotheses such as ADR, this a priori interesting tool proves to be fairly blunt: the linearly ordered sets are exactly those whose size is smaller than some P(κ) for an ordinal κ, and there is a smallest cardinal which is not linearly ordered, namely the E 0 cardinal [4]. However, if we insert an ultrafilter into our universe, the situation changes and linear orderability turns into a much more intriguing concept.…”

Section: Linear Orderabilitymentioning

confidence: 99%

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Canonical Ramsey Theory on Polish Spaces

Kanovei

Sabok

Zapletal

2013

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This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy–Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

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Section: Completely Regular Modelsmentioning

confidence: 99%

Section: Linear Orderabilitymentioning

confidence: 99%

Canonical Ramsey Theory on Polish Spaces

Kanovei

Sabok

Zapletal

2013

View full text Add to dashboard Cite

show abstract

“…Let us point out that Theorem 0.1 has a particularly nice form, if we assume Projective Determinacy or replace the Axiom of Choice with the Axiom of Determinacy (see, e.g. [CK11] for related results).…”

Section: Now Definementioning

confidence: 99%

On Homomorphism Graphs

Brandt,

Chang,

Grebík

et al. 2024

Forum of Mathematics, Pi

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We introduce new types of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks [Mar16]. The motivation for the construction comes from the adaptation of this method to the $\mathsf {LOCAL}$ model of distributed computing [BCG+21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, strengthening the main result of [TV21], we show that for $\Delta>2$ , it is impossible to give a simple characterization of acyclic $\Delta $ -regular Borel graphs with Borel chromatic number at most $\Delta $ : such graphs form a $\mathbf {\Sigma }^1_2$ -complete set. This implies a strong failure of Brooks-like theorems in the Borel context.

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On Supercompactness of $$\omega _1$$

Ikegami

Trang

2021

Springer Proceedings in Mathematics &Amp; Statistics

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A trichotomy theorem in natural models of 𝐴𝐷⁺

Cited by 14 publications

References 0 publications

Canonical Ramsey Theory on Polish Spaces

Canonical Ramsey Theory on Polish Spaces

On Homomorphism Graphs

On Supercompactness of $$\omega _1$$

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