**March 4**

Tom Benhamou,
Tel Aviv University

**Subforcings of the Tree-Prikry Forcing**

We investigate which forcing notions can be embedded into a Tree-Prikry forcing. It turns out that the answer changes drastically under different large cardinal assumptions. We will focus on the class of $\kappa\text{-}$strategically closed forcings of cardinality $\kappa$, $<\kappa\text{-}$strategically closed forcings of cardinality $\kappa$ and the $\kappa\text{-}$distributive forcing notions of cardinality $\kappa$. Then we will examine distributive subforcings of the Prikry forcing of cardinality larger than $\kappa$. This is a joint work with Moti Gitik and Yair Hayut.