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# Documents  03E75 | enregistrements trouvés : 8

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## Chain conditions, unbounded colorings and the $C$-sequence spectrum Rinot, Assaf | CIRM H

Post-edited

Research talks;Logic and Foundations

The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, was resolved by Shelah in 1997.
In the first part of this talk, we shall present analogous results regarding the infinite productivity of chain conditions stronger than $\kappa-cc$. In particular, for any successor cardinal $\kappa$, we produce a ZFC example of a poset with precaliber $\kappa$ whose $\omega ^{th}$ power is not $\kappa-cc$. To do so, we introduce and study the principle $U(\kappa , \mu , \theta , \chi )$ asserting the existence of a coloring $c:\left [ \kappa \right ]^{2}\rightarrow \theta$ satisfying a strong unboundedness condition.
In the second part of this talk, we shall introduce and study a new cardinal invariant $\chi \left ( \kappa \right )$ for a regular uncountable cardinal $\kappa$ . For inaccessible $\kappa$, $\chi \left ( \kappa \right )$ may be seen as a measure of how far away $\kappa$ is from being weakly compact. We shall prove that if $\chi \left ( \kappa \right )> 1$, then $\chi \left ( \kappa \right )=max(Cspec(\kappa ))$, where:
(1) Cspec$(\kappa)$ := {$\chi (\vec{C})\mid \vec{C}$ is a sequence over $\kappa$} $\setminus \omega$, and
(2) $\chi \left ( \vec{C} \right )$ is the least cardinal $\chi \leq \kappa$ such that there exist $\Delta\in\left [ \kappa \right ]^{\kappa }$ and
b : $\kappa \rightarrow \left [ \kappa \right ]^{\chi }$ with $\Delta \cap \alpha \subseteq \cup _{\beta \in b(\alpha )}C_{\beta }$ for every $\alpha < \kappa$.
We shall also prove that if $\chi (\kappa )=1$, then $\kappa$ is greatly Mahlo, prove the consistency (modulo the existence of a supercompact) of $\chi (\aleph_{\omega +1})=\aleph_{0}$, and carry a systematic study of the effect of square principles on the $C$-sequence spectrum.
In the last part of this talk, we shall unveil an unexpected connection between the two principles discussed in the previous parts, proving that, for infinite regular cardinals $\theta< \kappa ,\theta \in Cspec(\kappa )$ if there is a closed witness to $U_{(\kappa ,\kappa ,\theta ,\theta )}$.
This is joint work with Chris Lambie-Hanson.
The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, ...

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## Borel sets of Rado graphs are Ramsey Dobrinen, Natasha | CIRM H

Multi angle

Research talks;Combinatorics;Logic and Foundations

The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have attained such a result for Borel colorings of copies of the Rado graph. We build a topological space of copies of the Rado graph, forming a subspace of the Baire space. Using techniques developed for our work on the big Ramsey degrees of the Henson graphs, we prove that Borel partitions of this space of Rado graphs are Ramsey. The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have ...

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## On the classification of polish metric spaces up to isometry Goa, Su ; Kechris, Alexander S. | American Mathematical Society 2003

Ouvrage

- 78 p.
ISBN 978-0-8218-3190-8

Memoirs of the american mathematical society , 0766

Localisation : Collection 1er étage

espace polonais # groupe polonais # espace de Urysohn # réductible de Borel # isométrie

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## Ramsey methods in analysis Argyros, Spiros A. ; Todorcevic, Stevo | Springer 2005

Ouvrage

- 257 p.
ISBN 978-3-7643-7264-4

Advances courses in mathematics CRM Barcelona

Localisation : Ouvrage RdC (ARGY)

ensemble linéaire normé # théorie de Ramsey # théoie des ensembles

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## Walks on ordinals and their characteristics Todorcevic, Stevo | Birkhäuser 2007

Ouvrage

- 324 p.
ISBN 978-3-7643-8528-6

Progress in mathematics , 0263

Localisation : Collection 1er étage

logique mathématique # arithmétique ordinale # arithmétique cardinale # application à la théorie des théories de Ramsey # ensembles partiellement ordonné # espace de Banach # séparabilité

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## Banach Spaces and Descriptive Set Theory : Selected Topics Dodos, Pandelis | Springer 2010

Ouvrage

- xi, 161 p.
ISBN 978-3-642-12152-4

Lecture notes in mathematics , 1993

Localisation : Collection 1er étage

espace linéaire normé # espace de banach # théorie descriptive des ensembles # théorie de Ramsey

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## The structure of the real line Bukovsky, Lev | Birkhäuser 2011

Ouvrage

- xiv; 536 p.
ISBN 978-3-0348-0005-1

Monografie matematyczne , 0071

Localisation : Ouvrage RdC (BUKO)

fonction réelle # topologue élémentaire # application à la théorie des ensembles

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## Combinatorial set theory:with a gentle introduction to forcing Halbeisen, Lorenz J. | Springer 2012

Ouvrage

- xvi; 453 p.
ISBN 978-1-4471-2172-5

Springer monographs in mathematics

Localisation : Ouvrage RdC (HALB)

logique mathématique # théorie combinatoire des ensembles # théorie axiomatique des ensembles # nombre ordinal # nombre cardinal # invariant cardinal # théorème de Ramsey # forcing itéré # forcing

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