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H 2 Chain conditions, unbounded colorings and the $C$-sequence spectrum

Auteurs : Rinot, Assaf (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... Chain conditions Todorcevic's conjecture Galvin's approach The coloring axiom Instances of the axiom U A new cardinal invariant The strength of productivity The C-sequence spectrum Conjectures about the spectrum

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 :
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

 Informations sur la Vidéo Langue : Anglais Date de publication : 14/10/2019 Date de captation : 23/09/2019 Collection : Research talks ; Logic and Foundations Format : MP4 (.mp4) - HD Durée : 00:48:32 Domaine : Logic and Foundations Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-09-23_Rinot.mp4 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, BobanDates : 23/09/2019 - 27/09/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2052.html Citation Data DOI : 10.24350/CIRM.V.19564303 Cite this video as: 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

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