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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 $([\omega]^\omega,\subseteq^*)$-generic filter), so it is not equivalent to Sacks, Silver or Miller forcing.

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  • $\begingroup$ Can one implement the fusion arguments? It seems delicate to enforce the strong splitting requirement... $\endgroup$ Commented Apr 6 at 22:55
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    $\begingroup$ @JoelDavidHamkins We can implement fusion arguments. Define $T \leq_n U$ iff $T \subseteq U$ and the first $n$ splitting levels of $T$ and $U$ coincide. In that case, if $T_0 \geq_0 T_1 \geq_1 T_2 \geq_2 T_3 \cdots$ then $\bigcap_{n<\omega} T_n$ is also a strong subtree. $\endgroup$ Commented Apr 6 at 23:54
  • $\begingroup$ Somewhat related, the splitting level requirement is common when defining versions of the classical tree forcings on uncountable/inaccessible cardinals (like Sacks forcing, etc). $\endgroup$ Commented Apr 7 at 1:16

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This forcing notion occurred in Rosłanowski's Mycielski ideals generated by uncountable systems (1993) (section 4) and Brendle's Strolling through paradise (1995) (section 3.3). Usually this forcing notion is denoted by $\Bbb U$, as far as I know, and could be called "uniform Sacks forcing" or "Sacks forcing with uniform levels". As a poset (not as a forcing notion, by minimality, as you have noted), it is an intermediate between Silver forcing and Sacks forcing, since Silver forcing can be seen as the forcing with strong trees $T$ with the property that if $s,t\in T$ and $|s|=|t|$, then $\mathrm{succ}(s,T)=\mathrm{succ}(t,T)$ (that is, not only uniformity of the splitting levels, but uniformity at every level). Thus, under the appropriate representation of the forcing notions, if $\Bbb V$ and $\Bbb S$ denote the Silver and Sacks forcing notions, we have $\Bbb V\subseteq\Bbb U\subseteq \Bbb S$.

Like Sacks forcing, $\Bbb U$ satisfies Axiom $\mathsf{A}$, it is an ${}^\omega\omega$-bounding forcing notion and it preserves $P$-points, which distinguishes it from Silver forcing in another way than minimality.

Both papers also study the Marczewski $\sigma$-ideal $u^0$, consisting of all $\Bbb U$-null sets (defined as those subsets $A\subseteq{}^\omega 2$ such that every $T\in\Bbb U$ can be strengthened to $T'\leq T$ such that $[T']\cap A=\varnothing$). If $v^0$ and $s^0$ are the similarly defined $\sigma$-ideals corresponding to Silver and Sacks forcing, then there has been quite a bit of research about the relations between $u^0$, $v^0$ and $s^0$, and closely related Mycielski ideals, like $\mathfrak C_2$. Here $\mathfrak C_2$ is defined as the set of all $A\subseteq{}^\omega 2$ for which the second player has a winning strategy in every game $G(A,Y)$ where $Y\subseteq \omega$ is infinite; here the game $G(A,Y)$ has the players take turns to choose values for a real $x\in{}^\omega 2$, with first player playing at stage $n$ if and only if $n\notin Y$, and the first player wins if and only if $x\in A$. For example, Rosłanowski notes that $\Bbb U$ densely embeds into $\mathrm{Borel}({}^\omega 2)/\mathfrak C_2$. Although you might expect that this makes $\mathfrak C_2$ and $u^0$ equal, these are actually quite different ideals. See also, for instance, a recent paper by Kuiper & Spinas Mycielski ideals and uniform trees (2022).

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