| Résume | Two-cardinal tree properties have been a central topic of study in combinatorial set theory over the last fifteen years. Such principles can be used to characterize strongly compact and supercompact cardinals among inaccessible cardinals but can also consistently hold at smaller cardinals. Of particular interest has been their effect on cardinal arithmetic. Notably, a pair of results due to Viale and Krueger, respectively, shows that the strongest of these principles, $\mathsf{ISP}(\kappa)$, implies that the singular cardinals hypothesis holds above $\kappa$. This raises the natural question of whether the same conclusion follows from weaker principles. Motivated by this question, we introduce some combinatorial principles that hold in all known models of two-cardinal tree properties and prove that they imply a strong form of the singular cardinals hypothesis. We also show that these principles can consistently hold globally. This is in contrast with tree properties, where the possibility of a global consistency result for even the classical tree property remains a major open question. The talk will focus on the broader picture and will be accessible to a general logic audience. |
