Ken’s colorful questions

We shall prove both items simultaneously. Given a space $E$ and an open coloring $W$ over it, using SOCA there is an uncountable $T\subseteq E$ that is homogeneous. If $T$ is $W$-connected, then $T^{\dagger}\cap W=T^{\dagger}$ which is clopen and open dense in $T^{\dagger}$. Otherwise, $T$ is $W$-free so $T^{\dagger}\cap W=\emptyset$ which is clopen and empty. ∎

It suffices to show that given any uncountable linearly ordered separable metric space $X$, the increasing open coloring

of $X^{2}$ can be reduced to a clopen coloring.

To see this, recursively construct $T=\{x_{\xi}=(x_{1}^{\xi},x^{\xi}_{2}):\xi<\omega_{1}\}$ as follows: Assume $x_{\xi}=(x_{1}^{\xi},x^{\xi}_{2})$ for $\xi<\alpha$ have already been chosen. Since $\alpha$ is countable $X^{2}\setminus\left(\bigcup_{\xi<\alpha}(X\times\{x_{2}^{\xi}\}\cup\{x_{1}^{\xi}\}\times X)\right)\neq\emptyset$. Take $x_{\alpha}$ to be any point in it.

Now, given two distinct $(x,y),(z,w)\in T$, we have that $x\neq y$ and $z\neq w$, hence, $((x,y),(z,w))$ is either in $W$ or in $\{((a,b),(c,d)):a<c\leftrightarrow b>d\}$, but both are open sets. Therefore, $T\cap W$ is clopen in $T$.

We start the proof from the Dense Reduction by noting the following:

To see this, let $W$ be a clopen coloring over $E$ and assume that $T\subseteq E$ is an uncountable set that reduces $W$ to open dense. Notice that, given $(a,b)\in T^{\dagger}\cap W^{c}$, as $T$ reduces $W$ to open dense, $T^{\dagger}\cap W\cap U\neq\emptyset$ for every open set $(a,b)\in U$. On the other hand, as $W$ is clopen in $E$, there is an open set $V$ such that $(a,b)\in V\subseteq W^{c}$. Clearly, these two assumptions are contradicting each other. Therefore, $T^{\dagger}\cap W^{c}=\emptyset$ making $T$ open-homogeneous. This concludes the proof of the fact.

Let $W$ be the increasing coloring on $A^{2}$ where $A$ is a $2$-entangled set. We know that $W$ can be reduced to a clopen coloring, so lets assume that it is clopen. Notice that for any $T\subseteq A^{2}$, $T^{\dagger}\cap W$ is clopen. Using our fact, the only way to get an open dense coloring or an empty coloring would be finding an uncountable homogeneous set, which does not exist when $A$ is a $2$-entangled set. Then the existance of a $2$-entangled set implies the failure of Dense Reduction. ∎

The fact that Clopen SOCA implies $\mathfrak{b}>\aleph_{1}$ was proved in [12].

Now, assume, towards a contradiction, that every open coloring can be reduced to a clopen coloring and $\mathfrak{b}=\aleph_{1}$. Take a $\mathfrak{b}$-scale and let $W$ be an open coloring such that

where $f\leq g$ means that for all $n\in\omega$, $f(n)\leq g(n)$. The above coloring has no uncountable homogeneous set when $\mathfrak{b}=\aleph_{1}$ (see [17]). Furthermore, using the theory of oscillation of Todorčević [17], in any cofinal (for $\mathfrak{b}=\aleph_{1}$, uncountable) set of a $\mathfrak{b}$-scale there are $f$ and $h$, such that $h\leq^{\ast}f$ but $h\not\leq f$, so there is a value $m$ where $h(m)>f(m)$, so $W^{c}$ is closed but not clopen. Now, since every uncountable subset of a $\mathfrak{b}$-scale is also a $\mathfrak{b}$-scale (when $\mathfrak{b}=\aleph_{1}$), $W$ can never be reduced to a clopen coloring. This contradicts our assumptions.

Finally, towards a contradiction, assume Dense SOCA and $\mathfrak{b}=\aleph_{1}$. Using a $\mathfrak{b}$-scale $B$ of size $\aleph_{1}$, we can eliminate countably many points to have that every open set is uncountable. Using the same coloring $W$ as above, every open set in $B^{\dagger}$ has a subset of the form

which is an open set. So $W^{c}$ is nowhere dense (nwd). Furthermore, by a theorem of Todorčević [17], every $\mathfrak{b}$-scale has two functions, $f$ and $g$ such that $f<g$. Then, this coloring is open dense but it has no uncountable homogeneous set. ∎

We already know that SOCA implies all the other statements. The equivalence of Clauses 2 and 3 with 1 have identical natural proofs: first reducing any open coloring to one of the special kind and then apply the corresponding weakening of SOCA to that special coloring. The rest follows directly from the following two observations:

Fact 2.2 is a direct consequence of Fact 2.1. To show Fact 2.3, let $W$ be an open coloring over $E$. Notice that $W\cup\mbox{int}(W^{c})$ is an open dense coloring over $E$, since its complement is $\partial W$.³³3$\partial W$ denotes the boundary of $W$. It is nowhere dense and it can be define either as $\overline{W}\cap\overline{W^{c}}$ or as $\overline{W}\setminus\mbox{int}(W)$

Using Dense SOCA over $W\cup\mbox{int}(W^{c})$ we either get a closed-homogeneous uncountable set $T$, in which case, $T^{\dagger}\cap(W\cup\mbox{int}(W^{c}))=\emptyset$, so $T^{\dagger}\cap W=\emptyset$ is clopen, or we get an uncountable set such that $T^{\dagger}\subseteq W\cup\mbox{int}(W^{c})$. Because both $W$ and $W^{c}$ are open in $T^{\dagger}$ we have that $T$ reduces $W$ to a clopen coloring. This concludes the proof of the facts. ∎

Let $E$ be a separable metric Baire space and let $W\subseteq E^{\dagger}$ be open and symmetric.

By definition, $\partial W$ is meager (nowhere dense) and, by the Kuratowski-Ulam Theorem, we have that the set

is meager.

We can recursively construct a sequence $\langle x_{\xi}:\xi\in\omega_{1}\rangle$ contained in $E\setminus T_{0}$ such that, given $\alpha<\beta$, $x_{\beta}\notin(\partial W)_{x_{\alpha}}$. Once we construct this sequence we will be done: by symmetry of $W$, $(e,e^{\prime})\in\partial W$ if and only if $(e^{\prime},e)\in\partial W$. So, the above sequence will have the property that, given $\alpha$ and $\beta$, $(x_{\alpha},x_{\beta})\notin\partial W$. In other words, there is an open set $O$ of $E^{\dagger}$ such that $(x_{\alpha},x_{\beta})\in O$ and either $O\cap W=\emptyset$ or $O\subseteq W$. Therefore, $\{x_{\xi}:\xi\in\omega_{1}\}^{\dagger}\cap W$ will be clopen in $\{x_{\xi}:\xi\in\omega_{1}\}^{\dagger}$.

For the construction, assume that we already selected $x_{\xi}$ for $\xi<\alpha$, for $\alpha<\omega_{1}$. Since $\alpha$ is countable, $E$ is a Baire space and $x_{\xi}\notin T_{0}$ we have that $E\setminus\left(\bigcup_{\xi<\alpha}(\partial W)_{x_{\xi}}\cup T_{0}\right)\neq\emptyset$. We just let $x_{\alpha}$ be an element of $E\setminus\left(\bigcup_{\xi<\alpha}(\partial W)_{x_{\xi}}\cup T_{0}\right)$.∎

If in the proof above we assume that $W$ is open dense, then $W^{c}=\partial W$. So, the proof above generates a $W$-connected set. ∎

Let $\{f_{\alpha}:\alpha<\mathfrak{c}\}$ be an enumeration of all continuous functions from $G_{\delta}$ subsets of $\mathds{R}$ to $\mathds{R}$ and let $\mathds{R}=\{x_{\alpha}:\alpha<\mathfrak{c}\}$ be an enumeration of the reals. Finally, let $\{N_{\alpha}:\alpha<\mathfrak{c}\}$ be an enumeration of all closed nwd sets.

We will construct the $2$-entangled Luzin set $\mathcal{S}\subseteq\mathds{R}$ by recursion. Assume that we already have $\mathcal{S}_{\alpha}=\{x_{\gamma_{\xi}}:\xi<\alpha\}$. Let

where

Notice that $B_{\alpha}$ is meager (it is countable) and $(\bigcup_{\xi<\alpha}N_{\xi}\cup B_{\alpha})$ is a countable union of nwd (by CH), so they do not cover $\mathds{R}$. This shows that $\mathds{R}\setminus(\bigcup_{\xi<\alpha}N_{\xi}\cup B_{\alpha})$ is non empty and $\gamma_{\alpha}$ is well defined.

It is clear from the construction, and from the assumption that $\mathfrak{c}=\aleph_{1}$, that $\mathcal{S}$ is a Luzin set. To show that $\mathcal{S}$ is $2$-entangled we follow the proof of Lemma 4.2 by Todorčević in [17]. Let $\{(x_{\alpha_{\xi}},x_{\beta_{\xi}}):\xi<\mathfrak{c}\}\subseteq\mathds{R}^{2}$ be a collection of size continuum of disjoint $2$-tuples of $\mathcal{S}$.

Let

We can assume that this set is of size continuum. If not, we can run the argument interchanging the roles of $\alpha_{\xi}$ and $\beta_{\xi}$.

Now, we can define the function $g:K\rightarrow\mathds{R}$ such that

Furthermore, define the set

where⁴⁴4Here $B_{\frac{1}{n}}^{K}(s)$ is the ball of radius $\frac{1}{n}$ with center $s$ in $K$, a subset of $\mathds{R}$. $\omega_{g}(s)=\bigcap_{n\in\omega}\overline{g[B_{\frac{1}{n}}^{K}(s)]}$ is the oscillation of $g$ at $s$, i.e., all the accumulation points of the images (under $g$) of sequences that converge to $s$. Notice that $|\omega_{g}(s)|=1$ if and only if $g$ is continuous at $s$.

Recall that any partial continouos function from $\mathds{R}^{n}$ to $\mathds{R}$ can be extended to a partial function whose domain is a $G_{\delta}$ set. With this and our construction of $\mathcal{S}$ we have that the set $K_{0}$ is of size continuum.

Given $s\in K_{0}$, let $a_{s}$, $b_{s}$ be two distinct elements in $\omega_{g}(s)$. Without loss of generality, we can assume that $a_{s}<b_{s}$. Let $r\in\mathds{Q}$ such that $a_{s}<r<b_{s}$. Since we only have countably many rational numbers, we may shrink $K_{0}$ in such a way that for all $s\in K_{0}$ the rational number $r$ is the same. Notice that $K_{0}$ still has size continuum.

Take $t,s\in K_{0}$ such that $t<s$ and take disjoint intervals $I_{t},I_{s}$ such that $t\in I_{t}$ and $s\in I_{s}$. By the definition of $a_{t}$, $a_{s}$, $b_{t}$ and $b_{s}$ there are $t_{0},t_{1}\in K\cap I_{t}$ and $s_{0},s_{1}\in I_{s}\cap K$ such that $g(t_{0}),g(s_{0})<r<g(t_{1}),g(s_{1})$. Then for the pairs $(t_{0},g(t_{0}))$, $(s_{1},g(s_{1}))$ we have $t_{0}<s_{1}$ and $g(t_{0})<g(s_{1})$; and for the pair $(t_{1},g(t_{1}))$, $(s_{0},g(s_{0}))$ we have $t_{1}<s_{0}$ but $g(t_{1})>g(s_{0})$. ∎

Let

It is enough to show that this is a ccc poset. Once we have this, since it is of size $\kappa<\mathfrak{c}$, MA will imply that it is the union of countably many filters. Notice that the union of each filter is an open-homogeneous set and, since $\{e\}\in\mathds{C}_{E,W}$ for all $e\in E$, we have that $E$ is a countable union of open-homogeneous sets so, it has an open-homogeneous set of size $\kappa$ (using its cofinality).

In order to show that $\mathds{C}_{E,W}$ is ccc take $\{p_{\alpha}\ :\ \alpha<\omega_{1}\}\subseteq\mathds{C}_{E,W}$. We will show that there exist $\alpha\neq\beta$ such that $p_{\alpha}$ is compatible with $p_{\beta}$.

Since $|p_{\alpha}|\in\omega$ for all $\alpha$ we may assume that $\{p_{\alpha}\ :\ \alpha<\omega_{1}\}$ is a $\Delta$-system with root $r$. Furthermore, we may assume that $|p_{\alpha}\setminus r|=|p_{\beta}\setminus r|=m$. So, we can write $p_{\alpha}\setminus r=\{x^{0}_{\alpha},...,x^{m-1}_{\alpha}\}$

Let $\mathcal{B}$ be a countable basis of $E$. For each $\alpha$ we can choose $V_{i}\in\mathcal{B}$ such that for all $i\in m$, $x_{i}^{\alpha}\in V_{i}$ and for all $i\neq j$, $V_{i}\times V_{j}\subseteq W$ (this is possible since $W$ is open and for all $i\neq j$ and all $\gamma$ $(x^{\gamma}_{i},x^{\gamma}_{j})\in W$).

We will prove by induction on $m$ that there exists uncountable many elements that are compatible.

For $m=1$, let $T=\left\{x^{0}_{\alpha}:\alpha<\omega_{1}\right\}$. We know that $T$ do not have any uncountable closed-homogeneous set, so we can use SOCA($\aleph_{1}$) to choose an uncountable open-homogeneous set. Let $S\subseteq\omega_{1}$ be the indexes of the elements of this open-homogeneous set. Notice that all the elements of $\{p_{\alpha}\ :\ \alpha\in S\}$ are compatible.

Assume we have the result for $m$, we will prove it for $m+1$.

First, using the induction hypothesis, take $\{q_{\alpha}\ :\ \alpha<\omega_{1}\}$ such that $\{q_{\alpha}\setminus\{x_{\alpha}^{m}\}\ :\ \alpha<\omega_{1}\}$ are compatible. Now, let $T=\{x^{m}_{\alpha}:\alpha<\omega_{1}\}$. We know that $T$ do not have any uncountable closed-homogeneous set, so we can use SOCA($\aleph_{1}$) to choose an uncountable open-homogeneous set. Let $S\subseteq\omega_{1}$ be the indexes of the elements $x_{\alpha}^{m}$ in that uncountable open-homogeneous set. Notice that all the elements of $\{q_{\alpha}\ :\ \alpha\in S\}$ are compatible: first, all the elements of $\{q_{\alpha}\setminus\{x_{\alpha}^{m}\}\ :\ \alpha\in S\}$ are compatible; also, we have that $(x_{\alpha}^{m},x^{m}_{\beta})\in W$ for all $\alpha,\beta\in S$ and, finally, we have that $(x^{m}_{\alpha},x^{j}_{\beta})\in V_{m}\times V_{j}\subseteq W$ for all $\alpha,\beta\in\omega_{1}$ and all $j\in m$. ∎

The proof for this Lemma is inspired, indeed, by Todorčević and Moore [13] and by the proof of the consistency of SOCA with $\mathfrak{c}>\aleph_{2}$ in Abraham-Rudin-Shelah [1]. The inspiration of the former will be seen on the use of MTA. On the other hand, the strategy that Abraham-Rudin-Shelah used to prove their result was that given a metric space, they created a “tower of models” for each closed set of the metric space and its cartesian powers. After that, they “combined” all of them using their Axiom A1 (which, essentially, guarantees a fast club) and, finally, use the club method.

We will have a similar approach (even using the club method at the end). Nevertheless, we will change A1 for MTA and create “towers of models” for all the pairs of closed sets and an uncountable ordinal of $\omega_{2}$. We do this as follows:

Let $E=\{e_{\alpha}:\alpha<\omega_{2}\}$ and $\mathcal{B}=\{B_{n}:n\in\omega\}$ be a basis for its topology. Fix $\theta\geq\aleph_{3}$, and $b_{\beta}$ bijections between $\beta$ and $\omega_{1}$ for each $\beta\in\omega_{2}\setminus\omega_{1}$.

For each closed subset of $E^{n}$, say $F$, and each $\beta\in\omega_{2}\setminus\omega_{1}$ we define $\{M_{\gamma}^{F,\beta}:\gamma<\omega_{1}\}$ a continuous $\in$-chain of countable elementary submodels of $\langle H(\theta),E,\mathcal{B}\rangle$ such that given $\alpha<\beta$ we ensure that $e_{\alpha}\in M_{b_{\beta}(\alpha)+1}^{F,\beta}$. Whenever $F$ and $\beta$ are clear from the context we will use the notation $M_{\gamma}$ for $\gamma\in\omega_{1}$.

Notice that, since there are only $\aleph_{2}$ pairs $(F,\beta)$, we just define $\aleph_{2}$ “towers of models” each one of them with $\omega_{1}$ models such that for all $\gamma\in\omega_{1}$, $F\in M^{F,\beta}_{\gamma}$ and

Once again, for $F$ a closed subset of $E^{n}$ and $\beta\in\omega_{2}\setminus\omega_{1}$, we define $f_{F,\beta}:\omega_{1}\rightarrow\omega_{1}$ such that:

using the sequence of elementary submodels corresponding to $F$ and $\beta$. These functions are well define, since each $M_{\chi}$ is countable, and each of of them is a countable-to-one function.

The combinatorial principle MTA ensures the existance of a one-to-one function $g:\omega_{2}\rightarrow\omega_{2}$ satisfying the conditions of Lemma 4.5 for the set $\{f_{F,\beta}:\beta\in\omega_{2}\setminus\omega_{1},n\in\omega,F\subseteq E^{n}\}$.

Let $E_{0}=\{e_{g(\alpha)}:\alpha<\omega_{2}\}$. We claim that this set works.

To prove this, it is enough to show that given an uncountable collection of finite open-homogeneous sets, the union of two of them is open-homogeneous.

Assume that we have $\aleph_{1}$-many finite open-homogeneous sets, say $A=\{x_{\alpha}:\alpha<\omega_{1}\}$. We know that there is an uncountable $\beta<\omega_{2}$ such that for all $\alpha<\omega_{1}$, if $e_{\xi}\in x_{\alpha}$, then $\xi<\beta$. Furthermore, we can find this $\beta$ such that it is closed under $g$.

Without lost of generality, we can assume that they form a $\Delta$-system with empty root and they all have size $n$. As in Theorem 4.2, we can naturally associate a vector in $E^{n}$ to each $x_{\alpha}$. To do so, we enumerate $x_{\alpha}=\{e_{g(\xi^{1}_{\alpha})},...,e_{g(\xi^{n}_{\alpha})}\}$ in such a way that $i<j$ if and only if $b_{\beta}(g(\xi^{i}_{\alpha}))<b_{\beta}(g(\xi^{j}_{\beta}))$.

Shrinking $A$ if necessary, we can assume that, if $\delta<\gamma$, then

After these reductions, let $F$ be the closure of $\{(e_{g(\xi^{1}_{\alpha})},...,e_{g(\xi^{n}_{\alpha})}):\alpha<\omega_{1}\}$ in $E^{n}$.

We will work with the continuous $\in$-chain of models associated with $F$ and $\beta$.

Given $\alpha<\beta$, we say that the height of $e_{\alpha}$, denoted $ht^{F,\beta}(e_{\alpha})=ht(e_{\alpha})$, is the minimum $\gamma$ such that $e_{\alpha}\in M_{\gamma+1}\setminus M_{\gamma}$. Given our definition for these chains of models, we have that $ht(e_{\gamma})\leq b_{\beta}(\gamma)$ for $\gamma\in\beta$. On the other hand, the definition of $f_{F,\beta}$ ensure that, if $\alpha,\delta\in\omega_{1}$ and $f_{F,\beta}(\alpha)<\delta$ then $ht(e_{b^{-1}_{\beta}(\xi)})<\delta$ for all $\xi<\alpha$, $ht(e_{b^{-1}_{\beta}(\alpha)})<\delta$ and $\alpha<ht(e_{b_{\beta}^{-1}(\chi)})$ for all $\chi\geq\delta$.

We claim that, for all but countably many ordinals in $\beta$, if $b_{\beta}(g(\eta))<b_{\beta}(g(\xi))$ then $ht(e_{g(\eta)})<ht(e_{g(\xi)})$. To see this, from the second bullet of MTA, we have that there is a countable $D$ such that if $\eta\neq\xi\in\beta\setminus D$ and $b_{\beta}(g(\eta))<b_{\beta}(g(\xi))$ then $f_{F,\beta}(b_{\beta}(g(\eta))<b_{\beta}(g(\xi))$. This means that $b_{\beta}(g(\eta))<ht(e_{g(\xi)})$. Since $ht(e_{g(\eta)})\leq b_{\beta}(g(\eta))$, we have that $ht(e_{g(\eta)})<ht(e_{g(\xi)})$.

The above inequality allow us to find a really useful family of uncountable sets: given $\alpha<\omega_{1}$ such that $\xi_{\alpha}^{i}\in\beta\setminus D$ for all $i\in\{1,...,n\}$, let $\mu=ht(e_{g(\xi^{n}_{\alpha})})$ and $F_{\alpha}=\{a\in E\cap M_{\mu}:x_{\alpha}\mathord{\upharpoonright}_{1,...,n-1}\mathbin{\raisebox{4.30554pt}{\scalebox{0.7}{$\frown$}}}a\in F\}$. Since $b_{\beta}(g(\xi^{i}_{\alpha}))<b(g(\xi^{n}_{\alpha}))$ for $i<n$, we have that $ht(e_{g(\xi^{i}_{\alpha})})<ht(e_{g(\xi^{n}_{\alpha})})=\mu$. Then, $e_{g(\xi^{i}_{\alpha})}\in M_{\mu}$. Also, $F,x_{\alpha}\mathord{\upharpoonright}_{1,...,n-1}\in M_{\mu}$, which implies that $F_{\alpha}\in M_{\mu}$. This shows that $F_{\alpha}$ is uncountable. To see this, remember that given a countable set $L$, if $L\in M_{\mu}$ then $L\subseteq M_{\mu}$. Therefore, the fact that $F_{\alpha}\in M_{\mu}$, $\xi^{n}_{\alpha}\in F_{\alpha}$ but $\xi^{n}_{\alpha}\notin M_{\mu}$ shows that $F_{\alpha}$ is uncountable.

From here, it is enough to follow the exposition of the consistency of SOCA($\aleph_{1}$) as in [11] Lemmas V.6.14 and V.6.15. The technique presented there is the club method used in [1]. The club method has two steps: the preparation and the cloning. For the preparation, we select open sets for each $\alpha<\omega$ as we exposed here for Theorem 4.2.

Cloning resembles the technique in Theorem 3.5 where we use the oscillation of a discontinuous function, although the contexts are really different. The analogy comes from the fact that both techniques first find points in a set of accumulation points ($F$ and $F_{\alpha}$ here and the oscillation in Theorem 3.5) to fix open sets ($I_{s}$, $I_{r}$, $(-\infty,r)$ and $(r,\infty)$ in Theorem 3.5). Afterwards, both technique uses those open sets to find actual elements of the original uncountable set that are compatible (elements of $\{(e_{g(\xi^{1}_{\alpha})},...,e_{g(\xi^{n}_{\alpha})}):\alpha<\omega_{1}\}$ here and elements of $K$ in Theorem 3.5). ∎