English
Chain conditions, unbounded colorings and the $C$-sequence spectrum
Post-edited
Auteurs : Rinot, Assaf (Auteur de la Conférence)
CIRM (Editeur )
03E05 - Combinatorial set theory (logic)
03E35 - Consistency and independence results
03E75 - Applications
06E10 - Chain conditions, complete algebras
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2052/Slides/rinot--luminy2019.pdf
CIRM (Editeur )
Résumé :
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.
Keywords : knaster; precaliber; closed coloring; unbounded function
Codes MSC : 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.
Keywords : knaster; precaliber; closed coloring; unbounded function
03E05 - Combinatorial set theory (logic)
03E35 - Consistency and independence results
03E75 - Applications
06E10 - Chain conditions, complete algebras
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2052/Slides/rinot--luminy2019.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 14/10/2019 Date de captation : 23/09/2019 Sous collection : Research talks arXiv category : Logic Domaine : Logic and Foundations Format : MP4 (.mp4) - HD Durée : 00:48:32 Audience : Researchers Download : https://videos.cirm-math.fr/2019-09-23_Rinot.mp4 
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Informations sur la Rencontre
Nom de la rencontre : 15th International Luminy Workshop in Set Theory / XVe Atelier international de théorie des ensemblesOrganisateurs de la rencontre : Dzamonja, Mirna ; Velickovic, Boban Dates : 23/09/2019 - 27/09/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2052.html
Données de citation
DOI : 10.24350/CIRM.V.19564303Citer cette vidéo: Rinot, Assaf (2019). Chain conditions, unbounded colorings and the $C$-sequence spectrum. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19564303 URI : http://dx.doi.org/10.24350/CIRM.V.19564303 |
Voir aussi
- [Multi angle] Towards the right generalization of descriptive set theory to uncountable cardinals / Auteur de la Conférence Motto Ros, Luca.
- [Multi angle] Lower bounds for the perfect set property at weakly compact cardinals / Auteur de la Conférence Müller, Sandra.
- [Multi angle] Some results on set mappings / Auteur de la Conférence Komjáth, Péter.
- [Multi angle] Borel sets of Rado graphs are Ramsey / Auteur de la Conférence Dobrinen, Natasha.
Bibliographie
- LAMBIE-HANSON, Chris et RINOT, Assaf. Knaster and friends I: closed colorings and precalibers. Algebra universalis, 2018, vol. 79, no 4, p. 90. - https://doi.org/10.1007/s00012-018-0565-1
