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 $\kappa =\mathrm{\aleph }{}_2$, 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 ${\omega }^{th}$ 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 $c:{\left[\kappa \right]}^2\to \theta$ 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$(\kappa )$ := {$\chi (\overrightarrow{C})\mid \overrightarrow{C}$ is a sequence over $\kappa$} $\setminus \omega$, and
(2) $\chi \left(\overrightarrow{C}\right)$ is the least cardinal $\chi \le \kappa$ such that there exist $\mathrm{\Delta }\in {\left[\kappa \right]}^{\kappa }$ and
b : $\kappa \to {\left[\kappa \right]}^{\chi }$ with $\mathrm{\Delta }\cap \alpha \subseteq {\cup }_{\beta \in b(\alpha )}{C}_{\beta }$ 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 $\chi ({\mathrm{\aleph }}_{\omega +1})={\mathrm{\aleph }}_0$, 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 $\theta <\kappa ,\theta \in Cspec(\kappa )$ 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