Domatic partitions with the property of Baire

For $n<\omega$, let $D_n\subseteq 2^\omega$ contain all infinite binary strings of the form $0^{n\frown}1^\frown\ldots$. Thus $\{D_n\}_{n<\omega}$ form a family of $\omega$-many disjoint clopen total dominating sets. The theorem holds as we expand this family into a full partition of $2^\omega$.

The first part of the Theorem is essentially Theorem 1, where we leverage that $2^\omega$ with graph $G_1$ can always be seen as the hypercube graph on $2^n$, but every element carries a tail of junk information.

For the second part, define $\phi:2^\omega\to \omega$, such that for $x\in 2^\omega$, if $x_0$ is the unique element in $A\cap[x]_{\E_0}$, then $\phi(x)$ is the minimal $n<\omega$ such that for all $m\ge n$, $x(m)=x_0(m)$. Next define $\pi:\omega\to\omega$ such that for every $i<\omega$, $\pi^{-1}\{i\}\subseteq\omega$ is infinite.

Note that for every $x\in 2^\omega$ and $m\ge \phi(x)$, if $y$ is the result of flipping the $m$-th bit of $x$, then $(x,y)\in G_1$ and $\phi(y)=m+1$. Thus $x$ has $G_1$-neighbors of all sufficiently large $\phi(y)$. Consequently $x$ has $G_1$-neighbors of every possible $\pi\circ\phi(y)\in\omega$. Letting $D_i=(\pi\circ\phi)^{-1}\{i\}$, we see that $\{D_i\}_{i<\omega}$ is an $\omega$-part total domatic partition of $G_1$.

We will demonstrate the idea behind the proof using the example of $d=2$, for the fear that the more general case will make notations too cumbersome.

The idea behind the proof is as follows. There is a relatively obvious reason for why $G_d$ does not admit any infinite-part open domatic partition, which is that many vertices will witness the failure of such a domatic partition. In the BP case, there are still enough generic vertices to witness the failure of a domatic partition, in a very similar way.

Let $\{D_k\}_{k<\omega}$ be an $\omega$-part BP domatic partition of $G_2$. For $i\ne j$ finite, let $\gamma_{i,j}$ denote the automorphism of $2^\omega$ by flipping the two bits at locations $i,j$, and similarly define $\gamma_i$ as the automorphism that flips the one bit at $i$. Note that every $D_k$ is non-meager, as $D_k$ and it countably many copies under $\gamma_i$ and $\gamma_{i,j}$ cover all of $2^\omega$.

By localization, assume that $D_0$ is comeager in some basic open $N_s$. Thus any $\gamma_i$ or $\gamma_{i,j}$ with $i,j\ge |s|$ will be an automorphism of $N_s$. The intersection of $D_0$ with its copies under every such $\gamma_i$ and $\gamma_{i,j}$ will remain comeager in $N_s$, with the property that for any $x\in N_s$ in this intersection, any $G_2$-neighbor of the form $\gamma_i(x)$ or $\gamma_{i,j}(x)$ with $i,j\ge |s|$ will remain in $D_0$. Thus anytime this $x$ sees a $G_2$-neighbor that is not in $D_0$, some $<|s|$-th bit of $x$ must be flipped.

This leaves us with $|s|$-many cases. In particular, flipping any one bit of $x$ will make $x$ see at most $|s|$-many new colors, but $x$ will need to see all $\omega$-many colors, so a second bit must be flipped. If we can find some $x$ such that it sees only finitely many new colors even after flipping two bits, then we are done as $x$ is a witness of the failure of the domatic partition.

We will find sequential extensions $(s_n)_{n\le |s|}$ of $s$ as follows. Let $s_0=s$. For every $n<|s|$, consider the basic open set $N_{\gamma_n(s_n)}$. As $\{D_k\}_{k<\omega}$ partitions into it, there is some $k_n<\omega$ such that $D_{k_n}$ is non-meager in $N_{\gamma_n(s_n)}$. By localization, there is some $s_{n+1}\supseteq s_n$ such that $D_{k_n}$ is comeager in $N_{\gamma_n(s_{n+1})}$. By a similar discussion as before, for any generic $x\in N_{\gamma_n(s_{n+1})}$, the $G_1$-neighbors of $x$ will have only finitely many colors. Thus for any generic $x\in N_{s_{n+1}}$, the $G_1$-neighbors of $\gamma_n(x)$ will only have finitely many neighbors.

Thus as $s_{|s|}$ is a common extension of all $s_n$, any $x$ generic in $N_{s_{|s|}}$ will also be generic in every $N_{s_n}$, and we fix such an $x$. Thus for any $n<|s|$, $\gamma_n(x)$ has only finitely many $G_1$-neighbors. Now $\gamma_{i,j}(x)$ takes only one value when $i,j\ge |s|$, and $\gamma_{i,j}(x)$ takes on finitely many values when $i<|s|$ or $j<|s|$. This $x$ has the desired property of having only finitely many $G_2$-neighboring colors, and we arrived at a contradiction.

For the general case of $G_d$, we expect some form of "inductions inside inductions", as we iterate through all cases of how some $x$ gets connected to its $G_d$-neighbors. We still eventually stop after a finite amount of inductions, so the desired result holds.