Forcing with strong binary trees – mathoverflow.net

A strong subtree $T \subseteq 2^{<\omega}$ is a perfect tree of infinite height such that for all $s,t \in T$ for which $|s| = |t|$, $s$ splits in $T$ iff $t$ splits in $T$. The Ramsey theory of strong subtrees was studied by Milliken in A Partition Theorem for the Infinite Subtrees of a Tree. Let $\mathcal{M}$ denote the set of all strong subtrees. What is known about the forcing $(\mathcal{M},\subseteq)$? Note that $(\mathcal{M},\subseteq)$ is not minimal (e.g. it adds a...