I gave an invited talk at the 15th International Workshop on Set Theory in Luminy in Marseille, September 2019.
Talk Title: Chain conditions, unbounded colorings and the C-sequence spectrum.
Abstract: 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 $\omega^{\mathrm{th}}$ power is not $\kappa$-cc. To do so, we introduce and study the principle $\textrm{U}(\kappa,\mu,\theta,\chi)$ asserting the existence of a coloring $c:[\kappa]^2\rightarrow\theta$ satisfying a strong unboundedness condition.
In the second part of this talk, we shall introduce and study a new cardinal invariant $\chi(\kappa)$ for a regular uncountable cardinal $\kappa$. For inaccessible $\kappa$, $\chi(\kappa)$ may be seen as a measure of how far away $\kappa$ is from being weakly compact. We shall prove that if $\chi(\kappa)>1$, then $\chi(\kappa)=\max(\mathrm{Cspec}(\kappa))$, where:
1. $\textrm{Cspec}(\kappa) := \{\chi(\vec{C}) \mid \vec{C}$ is a $C$-sequence over $\kappa\} \setminus \omega$, and
2. $\chi(\vec{C})$ is the least cardinal $\chi \leq \kappa$ such that there exist $\Delta \in [\kappa]^\kappa$ and $b:\kappa\rightarrow[\kappa]^\chi$ with $\Delta\cap\alpha\subseteq\bigcup_{\beta\in b(\alpha)}C_\beta$ 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 $\theta<\kappa$, $\theta\in\mathrm{Cspec}(\kappa)$ iff there is a closed witness to $\mathrm{U}(\kappa,\kappa,\theta,\theta)$.
This is joint work with Chris Lambie-Hanson.
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