**November 4**

**Corey Switzer**,
University of Vienna

**The Special Tree Number**

A tree of height $\omega_1$ with no cofinal branch is called * special* if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function $f:T \to \omega$ so that if $f(s) = f(t)$ then $s$ and $t$ are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, $\mathfrak{st}$, the least size of a tree of height $\omega_1$ which is neither non-special nor has a cofinal branch. Classical facts imply that $\mathfrak{st}$ can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that $\mathfrak{st}$ can be larger than $\mathfrak{a}$, $\mathfrak{g}$, and both the left hand side and bottom row of the Cichon diagram. Thus $\mathfrak{st}$ is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.