A preservation theorem for theories without the tree property of the first kind

We prove that the NTP$_1$ property of a geometric theory $T$ is inherited by theories of lovely pairs and $H$-structures associated to $T$. We also provide a class of examples of nonsimple geometric NTP$_1$ theories

Club guessing and the universal models

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element