Abstract

The tree propperty at $kappa$ says that every tree of height $kappa$ and levels of size less than $kappa$ has a cofinal branch. A long term project in set theory is to get the consistency of the tree property at every regular cardinal greater than $leph_1$. So far we only know that it is possible to have the tree property up to $leph_{omega+1}$, due to Neeman. The next big hurdle is to obtain it both at $leph_{omega+1}$ and $leph_{omega+2}$ when $leph_omega$ is trong limit. Doing so would require violating the singular cardinal hypothesis at $leph_omega$.

In this tutorial we will start with some classic facts about the tree property, focusing on branch lemmas, successors of singulars and Prikry type forcing used to negate SCH . We will then go over recent developments including a dichotomy theorem about which forcing posets are good candidates for getting the tree property at $leph_{omega+1}$ together with not SCH at $leph_omega$. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously.