The 15th International Workshop on Set Theory in Luminy, September 2019

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 $\kappa ={\mathrm{\aleph }}_2$, 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 $\text{U}(\kappa ,\mu ,\theta ,\chi )$ asserting the existence of a coloring $c:[\kappa {]}^2\to \theta$ 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. $\text{Cspec}(\kappa ):=\{\chi (\overrightarrow{C})\mid \overrightarrow{C}$ is a $C$-sequence over $\kappa \}\setminus \omega$, and

2. $\chi (\overrightarrow{C})$ is the least cardinal $\chi \le \kappa$ such that there exist $\mathrm{\Delta }\in [\kappa {]}^{\kappa }$ and $b:\kappa \to [\kappa {]}^{\chi }$ with $\mathrm{\Delta }\cap \alpha \subseteq \bigcup _{\beta \in b(\alpha )}{C}_{\beta }$ 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 $\chi ({\mathrm{\aleph }}_{\omega +1})={\mathrm{\aleph }}_0$, 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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