Proof Assume $\mathrm {SAT}(\aleph _1)$ and $\neg \mathrm {CH}$ . Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _1)$ . Let $\mathcal {A}_\ast = (\omega _1, <)^\omega / \mathcal {U}$ . For $\alpha < \omega _1$ , put $\alpha _\ast = [{\langle \alpha , \alpha , \alpha , \dots \rangle }]$ . Define a set p of formulas with a free variable x by

$$\begin{align*}p = \{ \ulcorner \alpha_\ast < x \urcorner : \alpha < \omega_1 \}. \end{align*}$$

This p is finitely satisfiable and the number of parameters occurring in p is $\aleph _1 < \mathfrak {c} = \left \lvert {\mathcal {A}_\ast } \right \rvert $ by $\neg \mathrm {CH}$ . Thus, by $\mathrm {SAT}(\aleph _1)$ , we can take $f \colon \omega \to \omega _1$ such that $[f]$ realizes p. Put $\beta = \sup _{n \in \omega } f(n)$ . Now we have $\{ n \in \omega : \beta < f(n) \} \in \mathcal {U}$ and this contradicts the definition of $\beta $ .

Proof Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _2)$ . Let $\mathcal {A}_\ast = (\omega _2, <)^\omega / \mathcal {U}$ . For $\alpha < \omega _1$ , put $\alpha _\ast = [{\langle \alpha , \alpha , \alpha , \dots \rangle }]$ . Define a set p of formulas with a free variable x by

$$\begin{align*}p = \{ \ulcorner \alpha_\ast < x < (\omega_1)_\ast \urcorner : \alpha < \omega_1 \}. \end{align*}$$

The remaining argument is the same as Theorem 2.1.

Proof Take an increasing cofinal sequence ${\langle x_\alpha : \alpha < \omega _1\rangle }$ of points in $\mathcal {A}$ below a. Then ${\langle x_\alpha ^* : \alpha < \omega _1\rangle }$ is an increasing cofinal sequence in $\mathcal {A}_\ast $ , where $x_\alpha ^* = [{\langle x_\alpha , x_\alpha , x_\alpha , \dots \rangle }]$ for each $\alpha < \omega _1$ . This can be shown by regularity of $\omega _1$ .

Proof Consider the map $F \colon \mathbb {Q}^3 \to \mathbb {Q}$ defined by $F(x, y, z) = x - y + z$ . Then we have

$$\begin{align*}(\forall y, z \in \mathbb{Q}) (\text{the map } x \mapsto F(x, y, z) \text{ is an automorphism on } (\mathbb{Q}, <) \text{ that sends }y\text{ to }z). \end{align*}$$

This statement can be written by a first-order formula in the language $\mathcal {L}' = \{<, F\}$ . Thus the same statement is true in $(\mathbb {Q}, <, F)^\omega /\mathcal {U}$ . The map $F_\ast : \mathcal {B}_\ast ^3 \to \mathcal {B}_\ast $ induced by F satisfies that

$$\begin{align*}&(\forall y, z \in \mathcal{B}_\ast) \\&\quad(\text{the map } x \mapsto F(x, y, z) \text{ is an automorphism on } (\mathcal{B}_\ast, <) \text{ that sends }y\text{ to }z).\\[-35pt] \end{align*}$$

Proof This proof is based on [Reference Golshani and Shelah6, Theorem 2.1]. Assume that $\mathfrak {mcf} \ge \aleph _2$ . We shall show $\neg \operatorname {KT}(\aleph _1)$ .

Let $\mathcal {L} = \{ < \}$ , $\mathcal {A} = (\mathbb {Q}, <)$ and $\mathcal {B} = (\mathbb {Q} + ((\omega _1+1) \times \mathbb {Q}_{\ge 0}), <_{\mathcal {B}})$ . Here $<_{\mathcal {B}}$ is defined by a lexicographical order and a disjoint union order. $\mathcal {A}$ and $\mathcal {B}$ are dense linear ordered sets, so by completeness of DLO, we have $\mathcal {A} \equiv \mathcal {B}$ . Take an ultrafilter $\mathcal {U}$ over $\omega $ . Put $\mathcal {A}_* = \mathcal {A}^\omega / \mathcal {U}, \mathcal {B}_* = \mathcal {B}^\omega / \mathcal {U}$ .

There is a point a in $\mathcal {B}$ such that $\operatorname {cf}(\mathcal {B}_a) = \aleph _1$ , where $\mathcal {B}_a = \{ x \in \mathcal {B} : x < a \}$ . Then $a_* \in \mathcal {B}_*$ has cofinality $\aleph _1$ by Lemma 2.4. Here $a_* = [{\langle a, a, a, \dots \rangle }]$ . On the other hand, we shall show every point in $\mathcal {A}_*$ has cofinality $\ge \mathfrak {mcf}$ . If we do this, since we assumed $\mathfrak {mcf} \ge \aleph _2$ , we will have $\mathcal {A}_* \not \simeq \mathcal {B}_*$ .

By Lemma 2.5, it suffices to consider the point $0_* = [{\langle 0, 0, 0, \dots \rangle }]$ . Since $\mathbb {Q}$ is symmetrical, we consider $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}})$ .

Now we construct a Galois–Tukey morphism $(\varphi , \psi ) \colon \mathbf {Cof}(\omega ^\omega /\mathcal {U}) \kern1.2pt{\to}\kern1.2pt \mathbf {Cof}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}})$ by

$$ \begin{align*} \varphi &\colon \omega^\omega/\mathcal{U} \to (\mathbb{Q}_{>0})^\omega/\mathcal{U}; [f] \mapsto [{\langle1/(f(n)+1) : n \in \omega\rangle}], \\ \psi &\colon (\mathbb{Q}_{>0})^\omega/\mathcal{U} \to \omega^\omega/\mathcal{U}; [g] \mapsto [{\langle \left\lfloor {1/g(n)-1} \right\rfloor : n \in \omega\rangle}]. \end{align*} $$

So we have $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}}) \ge \operatorname {cf}(\omega ^\omega /\mathcal {U}, <_{\mathcal {U}})$ .

Thus we have $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}}) \ge \mathfrak {mcf}$ . We are done.

Proof This follows from Theorem 2.6 and the fact that $\mathfrak {b} \le \mathfrak {mcf}$ .

Fact 3.1 [Reference Blass4, Theorem 7.13]

The statement $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c}$ is equivalent to $\operatorname {MA}(\textrm {countable})$ , that is for every countable poset $\mathbb {P}$ and a family of dense sets $\mathcal {D}$ with $ \left \lvert {\mathcal {D}} \right \rvert < \mathfrak {c}$ there is a filter G of $\mathbb {P}$ that intersects all $D \in \mathcal {D}$ .

Proof [Reference Golshani and Shelah6, Theorem 3.3] shows that $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land \operatorname {cf}(\mathfrak {c})> \aleph _1$ implies $\operatorname {KT}(\aleph _1)$ and the exact same proof works for $\operatorname {KT}(\aleph _0)$ without the assumption $\operatorname {cf}(\mathfrak {c})> \aleph _1$ .

Here we sketch the proof.

Let $\mathcal {L}$ be a countable language and $\mathcal {A}^0$ and $\mathcal {A}^1$ are countable $\mathcal {L}$ -structures which are elementarily equivalent.

Enumerate $(\mathcal {A}^i)^\omega $ for $i=0, 1$ as

$$ \begin{align*} (\mathcal{A}^i)^\omega = \{f^i_\alpha : \alpha < \mathfrak{c} \}. \end{align*} $$

By a back-and-forth method, we construct a sequence of triples ${\langle (\mathcal {U}_\alpha , g^0_\alpha , g^1_\alpha ) : \alpha < \mathfrak {c}\rangle }$ satisfying:

1. (1) $g^0_\alpha \in \mathcal {A}^0$ .

2. (2) $g^1_\alpha \in \mathcal {A}^1$ .

3. (3) $\mathcal {U}_\alpha $ is a filter over $\omega $ generated by $\aleph _0 + \left \lvert {\alpha } \right \rvert $ sets.

4. (4) $(\mathcal {U}_\alpha : \alpha < \mathfrak {c})$ is an increasing continuous sequence.

5. (5) If $\varphi (x_0, \dots , x_{n-1}$ is an $\mathcal {L}$ -formula and $\beta _0, \dots , \beta _n \le \alpha $ , then the set

   $$\begin{align*}\{k \in \omega : \mathcal{M}^0 \models \varphi(g^0_{\beta_0}(k), \dots, g^0_{\beta_{n-1}}(k)) \iff \mathcal{M}^1 \models \varphi(g^1_{\beta_0}(k), \dots, g^1_{\beta_{n-1}}(k)) \}\end{align*}$$

   belongs to $\mathcal {U}_{\alpha +1}$ .

In the construction, when $\alpha $ is even, we put $g^0_\alpha = f^0_\gamma $ where $\gamma $ is the least ordinal $f^0_\gamma \not \in \{g^0_\beta : \beta < \alpha \}$ . And $\mathbb {P}$ is the poset of finite partial functions from $\omega $ to $\mathcal {A}^1$ . Take a generating set $\mathcal {F}$ of $\mathcal {U}_\alpha $ of size $\aleph _0 + \left \lvert {\alpha } \right \rvert $ . Then by using $\operatorname {MA}(\textrm {countable})$ , take a $\mathbb {P}$ -generic filter G with respect to the following family of dense sets of $\mathbb {P}$ :

$$\begin{align*}D_n = \{ p \in \mathbb{P} : n \in \operatorname{dom} p \} \text{ (for }n \in \omega) \end{align*}$$

and

$$ \begin{align*} E_{X,{\langle\varphi_\iota : \iota \in I\rangle},{\langle\gamma^\iota_1,\dots,\gamma^\iota_{n_\iota} : \iota \in I\rangle}} = &\{ p \in \mathbb{P} : (\exists k \in \operatorname{dom}(p) \cap X)(\forall \iota \in I) \\ & \hspace{0.4cm} (M^0 \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\alpha(k)) \Leftrightarrow \\ & \hspace{0.54cm} M^1 \models \varphi_\iota(g^1_{\gamma^\iota_1}(k), \dots g^1_{\gamma^\iota_{n_\iota}}(k), p(k)) \}), \end{align*} $$

where $X \in \mathcal {F}$ , ${\langle \varphi _\iota : \iota \in I\rangle }$ is a finite sequence of $\mathcal {L}$ -formulas and $\gamma ^\iota _1, \dots , \gamma ^\iota _{n_\iota }$ for $\iota \in I$ are ordinals less than $\alpha $ . Then putting $g^1_\alpha = \bigcup G$ satisfies the induction hypothesis.

Then the appropriate construction guarantees that $\mathcal {U} = \bigcup _{\alpha < \mathfrak {c}} \mathcal {U}_\alpha $ is an ultrafilter and that the function

$$\begin{align*}{\langle([g^0_\alpha]_{\mathcal{U}}, [g^1_\alpha]_{\mathcal{U}}) : \alpha < \mathfrak{c}\rangle} \end{align*}$$

is isomorphic from $(M^0)^\omega /\mathcal {U}$ to $(M^1)^\omega /\mathcal {U}$ .

Proof $\mathrm {MA} + \neg \mathrm {CH}$ implies $\operatorname {KT}(\aleph _0) \land \neg \operatorname {KT}(\aleph _1)$ by Theorems 2.6 and 3.2.

Fact 3.4 [Reference Bartoszyński and Judah2, Lemma 2.4.2]

$\operatorname {cov}(\mathsf {meager}) = \mathfrak {v}^\exists _{{\langle \omega : n \in \omega \rangle }, \mathrm {id}}$ . In other words, $\operatorname {cov}(\mathsf {meager}) \ge \kappa $ holds iff $(\forall X \subseteq \omega ^\omega \text { of size } <\kappa ) (\exists S \in \prod _{i \in \omega } [\omega ]^{\le i}) (\forall x \in X) (\exists ^\infty n) (x(n) \in S(n))$ holds.

Proof Take an ultrafilter $\mathcal {U}$ that witnesses $\mathrm {SAT}(\aleph _0)$ . Fix $X \subseteq \omega ^\omega $ of size $< \mathfrak {c}$ . Define a language $\mathcal {L}$ by $\mathcal {L} = \{\subseteq \}$ and for each $i \in \omega $ , define an $\mathcal {L}$ -structure $\mathcal {A}_i$ by $\mathcal {A}_i = ([\omega ]^{\le i}, \subseteq )$ . For each $x \in \omega ^\omega $ , let $S_x = {\langle \{x(i)\} : i \in \omega \rangle }$ . In the ultraproduct $\mathcal {A}_* = \prod _{i\in \omega } \mathcal {A}_i / \mathcal {U}$ , define a set p of formulas of one free variable S by

$$\begin{align*}p = \{ \ulcorner [S_x] \subseteq S \urcorner : x \in X \}. \end{align*}$$

This p is finitely satisfiable. In order to check this, let $x_0, \dots , x_n$ be finitely many members of X. Define S by $S(m) = \{ x_0(m), \dots , x_n(m) \}$ for $m \ge n$ . We don’t need to care about $S(m)$ for $m < n$ . Then this S satisfies $[S_{x_i}] \subseteq [S]$ for all $i \le n$ . Moreover, the number of parameters of p is $< \mathfrak {c}$ .

So by $\mathrm {SAT}(\aleph _0)$ , we can take $[S] \in \mathcal {A}_*$ that realizes p. Then S fulfills $(\forall x \in X) (\{ n \in \omega : x(n) \in S(n) \} \in \mathcal {U})$ . Thus $(\forall x \in X) (\exists ^\infty n) (x(n) \in S(n))$ .

Proof Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _0)$ . Fix $\kappa < \mathfrak {c}$ .

Put $\mathcal {L} = \{\subseteq \}$ and define an $\mathcal {L}$ -structure $\mathcal {A}$ by $\mathcal {A} = ([\omega ]^{<\omega }, \subseteq )$ . Put $\mathcal {A}^* = \mathcal {A}^\omega /\mathcal {U}$ .

Define a map $\iota \colon \omega ^\omega /\mathcal {U} \to \mathcal {A}^*$ by $\iota ([x]) = [{\langle \{x(n)\} : n \in \omega \rangle }]$ . By Fact 1.8, we have $ \left \lvert {\omega ^\omega /\mathcal {U}} \right \rvert = \mathfrak {c}$ . Take a subset F of $\omega ^\omega /\mathcal {U}$ of size $\kappa $ .

For each $X \subseteq F$ , let $p_X$ be a set of formulas with a free variable z defined by

$$\begin{align*}p_X = \{ \ulcorner \iota(y) \subseteq z \urcorner : y \in X \} \cup \{ \ulcorner \iota(y) \not \subseteq z \urcorner : y \in F \smallsetminus X \}. \end{align*}$$

Each $p_X$ is finitely satisfiable. In order to check this, take $[x_0], \dots , [x_n] \in X$ and $[y_0], \dots , [y_m] \in F \smallsetminus X$ . Put $z(i) = \{ x_0(i), \dots , x_n(i) \}$ . Then $\iota ([x_0]), \dots , \iota ([x_n]) \subseteq _{\mathcal {U}} [z]$ . In order to prove $\iota ([y_j]) \not \subseteq _{\mathcal {U}} [z]$ for each $j \le m$ , suppose that $\{ i \in \omega : y_j(i) \in z(i) \} \in \mathcal {U}$ . Then for each $i \in \omega $ , there is a $k_i \le n$ such that $\{ i \in \omega : y_j(i) = x_{k_i}(i)\} \in \mathcal {U}$ . Then there is a $k \le n$ such that $\{ i \in \omega : y_j(i) = x_k(i)\} \in \mathcal {U}$ . This implies $[y_j] = [x_k]$ , which is a contradiction.

By $\mathrm {SAT}(\aleph _0)$ , for each $X \subseteq F$ , take $[z_X] \in \mathcal {A}^*$ that realizes $p_X$ . For $X, Y \subseteq F$ with $X \ne Y$ , we have $[z_X] \ne [z_Y]$ . So $2^\kappa = \left \lvert {\{[z_X] : X \subseteq F\}} \right \rvert \le \left \lvert {\mathcal {A}^*} \right \rvert = \mathfrak {c}$ . Therefore we have proved $2^{<\mathfrak {c}} = \mathfrak {c}$ .

Proof This proof is based on [Reference Ellentuck and Rucker5, Theorem 1].

Let ${\langle b_\alpha : \alpha < \mathfrak {c}\rangle }$ be an enumeration of $\omega ^\omega $ . Let ${\langle (\mathcal {L}_\xi , \mathcal {B}_\xi , \Delta _\xi ) : \xi < \mathfrak {c}\rangle }$ be an enumeration of triples $(\mathcal {L}, \mathcal {B}, \Delta )$ such that $\mathcal {L}$ is a countable language, $\mathcal {B} = {\langle \mathcal {A}_i : i \in \omega \rangle }$ is a sequence of $\mathcal {L}$ -structures with universe $\omega $ , and $\Delta $ is a subset of $\operatorname {Fml}(\mathcal {L}^+)$ with $ \left \lvert {\Delta } \right \rvert < \mathfrak {c}$ . Here $\mathcal {L}^+ = \mathcal {L} \cup \{ c_\alpha : \alpha < \mathfrak {c} \}$ where the $c_\alpha $ ’s are new constant symbols and $\operatorname {Fml}(\mathcal {L}^+)$ is the set of all $\mathcal {L}^+$ formulas with one free variable. Here we used the assumption $2^{<\mathfrak {c}} = \mathfrak {c}$ . And ensure each $(\mathcal {L}, \mathcal {B}, \Delta )$ occurs cofinally in this sequence.

For $\mathcal {B}_\xi = {\langle \mathcal {A}^\xi _i : i \in \omega \rangle }$ , put $\mathcal {B}_\xi (i) = (\mathcal {A}^\xi _i, b_0(i), b_1(i), \dots )$ , which is an $\mathcal {L}^+$ -structure.

Let ${\langle X_\xi : \xi < \mathfrak {c}\rangle }$ be an enumeration of $\mathcal {P}(\omega )$ .

We construct a sequence ${\langle F_\xi : \xi < \mathfrak {c}\rangle }$ of filters inductively so that the following properties hold:

1. (1) $F_0$ is the filter consisting of all cofinite subsets of $\omega $ .

2. (2) $F_\xi \subseteq F_{\xi + 1}$ and $F_\xi = \bigcup _{\alpha < \xi } F_\alpha $ for $\xi $ limit.

3. (3) $X_\xi \in F_{\xi +1}$ or $\omega \smallsetminus X_\xi \in F_{\xi +1}$ .

4. (4) $F_\xi $ is generated by $< \mathfrak {c}$ members.

5. (5) If

   (*) $$ \begin{align} \text{for all } \Gamma \subseteq \Delta_\xi \text{ finite, } \{i\in\omega : \Gamma \text{ is satisfiable in } \mathcal{B}_\xi(i) \} \in F_\xi, \end{align} $$

   then there is an $f \in \omega ^\omega $ such that for all $\varphi \in \Delta _\xi $ , $\{ i \in \omega : f(i) \text { satisfies } \varphi \text {in } \mathcal {B}_\xi (i) \} \in F_{\xi +1}$ .

Suppose we have constructed $F_\xi $ . We construct $F_{\xi +1}$ . Let $F_\xi '$ be a generating subset of $F_\xi $ with $ \left \lvert {F_\xi '} \right \rvert < \mathfrak {c}$ . If (*) is false, let $F_{\xi +1}$ be the filter generated by $F_\xi ' \cup \{X_\xi \}$ or $F_\xi ' \cup \{\omega \smallsetminus X_\xi \}$ . Suppose (*).

Put $\mathbb {P} = \operatorname {Fn}(\omega , \omega ) = \{ p : p\text { is a finite partial function from }\omega \text { to }\omega \}$ . For $n \in \omega $ , put

$$\begin{align*}D_n = \{ p \in \mathbb{P} : n \in \operatorname{dom} p \}.\end{align*}$$

For $A \in F_\xi '$ and $\varphi _1, \dots , \varphi _n \in \Delta _\xi $ , put

$$\begin{align*}E_{A,\varphi_1,\dots,\varphi_n} = \{ p \in \mathbb{P} : (\exists i \in \operatorname{dom} p \cap A) (p(i) \text{ satisfies } \varphi_1, \dots, \varphi_n \text{ in } \mathcal{B}_\xi(i)) \}. \end{align*}$$

Each $D_n$ is clearly dense. In order to show that each $E_{A,\varphi _1,\dots ,\varphi _n}$ is dense, take $p \in \mathbb {P}$ . By (*) and the property $A \in F_\xi $ , we can take $i \in A \smallsetminus \operatorname {dom} p$ and $k \in \omega $ such that k satisfies $\varphi _1, \dots , \varphi _n$ in $\mathcal {B}_\xi (i)$ . Put $q = p \cup \{ (i, k) \}$ . This is an extension of p in $E_{A,\varphi _1,\dots ,\varphi _n}$ .

By using $\operatorname {MA}(\textrm {countable})$ , take a generic filter $G \subseteq \mathbb {P}$ with respect to above dense sets. Put $f = \bigcup G$ . Then $F_\xi " := F_\xi ' \cup \{ Y_\varphi : \varphi \in \Delta _\xi \}$ satisfies finite intersection property, where $Y_\varphi = \{ i \in \omega : f(i) \text { satisfies } \varphi \text { in } \mathcal {B}_\xi (i) \}$ . In order to check this, let $A \in F_\xi '$ and $\varphi _1, \dots , \varphi _n \in \Delta _\xi $ . Then by genericity, we can take $p \in G \cap E_{A,\varphi _1,\dots ,\varphi _n}$ . So we can take $i \in \operatorname {dom} p \cap A$ such that $p(i) \text { satisfies } \varphi _1, \dots , \varphi _n \text { in } \mathcal {B}_\xi (i)$ . Then we have $i \in A \cap Y_{\varphi _1} \cap \dots \cap Y_{\varphi _n}$ .

Let $F_{\xi +1}$ be the filter generated by $F_\xi '' \cup \{X_\xi \}$ or $F_\xi '' \cup \{\omega \smallsetminus X_\xi \}$ .

We have constructed ${\langle F_\xi : \xi < \mathfrak {c}\rangle }$ . In order to check that the resulting ultrafilter $F = \bigcup _{\xi < \mathfrak {c}} F_\xi $ witnesses $\mathrm {SAT}(\aleph _0)$ , let $\mathcal {L}$ and $\mathcal {B} = {\langle \mathcal {A}_i : i \in \omega \rangle }$ satisfy the assumption of the theorem. Let $\Delta $ be a subset of $\operatorname {Fml}(\mathcal {L}^+)$ with $ \left \lvert {\Delta } \right \rvert < \mathfrak {c}$ . Assume that for all $\Gamma \subseteq \Delta $ finite, $X_\Gamma := \{i\in \omega : \Gamma \text { is satisfiable in } \mathcal {B}_\xi (i) \} \in F$ . By the regularity of $\mathfrak {c}$ , we have $\alpha < \mathfrak {c}$ such that for all $\Gamma \subseteq \Delta $ finite, $X_\Gamma \in F_\alpha $ . Let $\xi \ge \alpha $ be satisfying $(\mathcal {L}_\xi , \mathcal {B}_\xi , \Delta _\xi ) = (\mathcal {L}, \mathcal {B}, \Delta )$ . Then by (5), there is an $f \in \omega $ such that for all $\varphi \in \Delta $ , $\{ i \in \omega : f(i) \text { satisfies } \varphi \text { in } \mathcal {B}(i) \} \in F$ . Thus $\prod _{i \in \omega } \mathcal {A}_i / F$ is saturated.

Proof Suppose that $A \subseteq M$ is connected but not an antichain. Then we can find elements $a_0, \dots , a_n \in M$ such that

$$ \begin{align*} M \models &(a_0 E a_1 \lor a_1 E a_0) \land \dots \land (a_{n-1} E a_n \lor a_n E a_{n-1}) \land \\ &(a_0\text{ and }a_n\text{ are comparable with respect to }\triangleleft). \end{align*} $$

By elementarity, we have $n+1$ many elements in $\Gamma $ that satisfies the same formula. This is a contradiction. So every connected subset in M is an antichain.

Note that any two connected vertexes in $\Gamma $ have a path of length at most $4$ . Thus we have

$$ \begin{align*} \Gamma \models (\forall a, b) &((a\text{ and }b\text{ are incomparable with respect to }\triangleleft) \\ &\rightarrow (\text{there is a path between }a\text{ and }b\text{ with length at most } 4)). \end{align*} $$

By elementarity, the same formula holds in M. So every antichain in M is connected.

Therefore the connected components of M are precisely the maximal antichains of M with respect to $\triangleleft $ .

Proof Let $F = \{ u_1, \dots , u_n \}$ . Observe that

$$ \begin{align*} \Gamma \models (\forall x_1) \dots (\forall x_n) & [ x_1, \dots, x_n \text{ are points in } U \text{ and belong to} \\ & \text {the same connected component and } \\ &\text{ the index of this connected component is } \ge n \\ & \rightarrow (\exists y) [y \text{ belongs to this component, } y \in V \text{ and } x_1, \dots, x_n \mathrel{E} y]]. \end{align*} $$

By elementarity, $\Gamma _{\mathrm {NS}}$ satisfies the same formula.

Proof We may assume that each $U^{\Delta _n} = \omega $ . Let $\{ f_i : i < \mathfrak {d} \}$ be a cofinal subset of $(\omega ^\omega , <^*)$ . For each $n, m \in \omega $ , take $v_{n, m} \in V^{\Delta _n}$ that is connected with first m points in $U^{\Delta _n}$ . For $i < \mathfrak {d}$ , put

$$\begin{align*}v_i = [{\langle v_{n, f_i(n)} : n \in \omega\rangle}]. \end{align*}$$

Let $[u] \in U^R$ . Consider u as an element of $\omega ^\omega $ . Take $f_i$ that dominates u. Then we have

$$\begin{align*}\{ n \in \omega : u(n) \mathrel{E^{\Delta_n}} v_{n, f_i(n)} \} \in \mathcal{V}. \end{align*}$$

Therefore $[u] \mathrel {E^R} v_i$ .

Proof Fix a connected component $C_0$ of Q and $[x_0] \in C_0$ . Then for each $n \in \omega $ , there is an infinite component $C_n$ above $x_0(n)$ . Now

$$\begin{align*}C = \{ [x] \in Q : x \in (\Gamma_{\mathrm{NS}})^\omega \text{ and } (\forall n \in \omega)(x(n) \in C_n) \} \end{align*}$$

is a connected component of Q above $C_0$ . Since C can be viewed as $C = \prod _{n \in \omega } C_n / \mathcal {V}$ , the conclusion of the lemma holds for C by Lemmas 4.7 and 4.8.

Proof Let $f \colon \omega \to \Gamma $ satisfy $f(n) \in G_n$ for all n. Let $C_0$ be the connected component that $[f]$ belongs to. Take a connected component C such that $C_0 \triangleleft C$ and an element $[g] \in C$ . Take a function $h \colon \omega \to \omega $ such that $\{ n \in \omega : g(n) \in G_{h(n)} \} \in \mathcal {U}$ . Then $A := \{ n \in \omega : h(n) \ge n \} \in \mathcal {U}$ . Put $h'(n) = \max \{ h(n), n\}$ .

Take ${\langle [v_i] : i < \kappa \rangle }$ with each $[v_i] \in C \cap V^P$ . Then we have

$$\begin{align*}B_i := \{ n \in \omega : v_i(n) \in G_{h(n)} \cap V^\Gamma \} \in \mathcal{U}. \end{align*}$$

Take $v_i'$ such that $v_i'(n) = v_i(n)$ for $n \in A \cap B_i$ and $v_i'(n) \in [h'(n)^3]^{\le h'(n)}$ for $n\in \omega $ . The assumption $\kappa < \mathfrak {c}^\exists $ and the calculation

$$\begin{align*}\sum_{n \ge 1} \frac{h'(n)}{h'(n)^3} = \sum_{n \ge 1} \frac{1}{h'(n)^2} \le \sum_{n \ge 1} \frac{1}{n^2} < \infty \end{align*}$$

give an $x \in \prod h'$ such that for all $i < \kappa $ , $(\forall ^\infty n) (x(n) \not \in v_i'(n))$ . For each $i < \kappa $ , take $n_i$ such that $(\forall n \ge n_i) (x(n) \not \in v_i'(n))$ .

Take a point $[u] \in C \cap U^P$ such that $u(n) = x(n)$ for all $n \in A$ . Then for all $i < \kappa $ we have

$$\begin{align*}\{ n \in \omega : u(n) \not \mathrel{E^\Gamma} v_i(n) \} \supseteq A \cap B_i \cap [n_i, \omega) \in {\mathcal{U}}. \end{align*}$$

Therefore $[u] \not \mathrel{E^P} [v_i]$ for all $i < \kappa $ .

Fact 4.11 [Reference Klausner and Mejía8, Lemma 2.3]

$\operatorname {cov}(\mathsf {null}) \le \mathfrak {c}^\exists $ .

Proof This corollary holds since $\aleph _1 = \mathfrak {d} < \operatorname {cov}(\mathsf {null}) = \mathfrak {c}$ in the random model.

Proof This proof is based on [Reference Golshani and Shelah6, Theorem 3.3].

Let G be a $(V, \mathbb {P}_{\omega _1})$ -generic filter.

Let $\mathcal {L}$ be a countable language and $M^0 \equiv M^1$ be two $\mathcal {L}$ -structures of size $\le \aleph _1$ in $V[G]$ . Take sequences ${\langle M^l_i : i < \omega _1\rangle }$ for $l = 0, 1$ that are increasing and continuous such that each $M^l_i$ is countable elementary substructure of $M^l$ and $M^l = \bigcup _{i < \omega _1} M^l_i$ . We can take an increasing sequence ${\langle \alpha _i : i < \omega _1\rangle }$ of even ordinals such that $M^l_i \in V[G_{\alpha _i + 1}]$ for every $l < 2$ and $i < \omega _1$ .

For $i < \omega _1$ and $\beta < \lambda $ , let $c^i_\beta $ be the $\beta $ -th Cohen real added by $\dot {\mathbb {Q}}_{\alpha _i}$ .

Take an enumeration ${\langle X_\gamma : \gamma < \lambda \cdot \omega _1 \rangle }$ of $\mathcal {P}(\omega )$ such that ${\langle X_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ for every $i < \omega _1$ . We can take such a sequence. The reason for this is that we can take ${\langle \dot {X}_\gamma : \lambda \cdot i \le \gamma < \lambda \cdot (i+1)\rangle }$ as an enumeration of $\mathbb {P}_{\alpha _i+1}$ nice names for subsets of $\omega $ and put $X_\gamma = (\dot {X}_\gamma )^G$ .

For each $l < 2$ , take an enumeration ${\langle f^l_\gamma : \gamma < \lambda \cdot \omega _1\rangle }$ of $(M^l)^\omega $ such that $f^l_{\lambda \cdot i + \beta } \in (M^l_i)^\omega $ for every $i < \omega _1$ and $\beta < \lambda $ and ${\langle f^l_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ .

For $\lambda ' < \lambda $ , let $G_{\alpha _i, \lambda '}$ denote $G \cap (\mathbb {P}_{\alpha _i} \ast \mathbb {C}_{\lambda '})$ .

Now we construct a sequence of quadruples ${\langle (\mathcal {U}_\gamma , g^0_\gamma , g^1_\gamma , \lambda _\gamma ) : \gamma < \lambda \cdot \omega _1\rangle }$ by induction so that the following properties hold.

1. (1) Each $\mathcal {U}_\gamma $ is a filter over $\omega $ .

2. (2) For every $l < 2$ , $i < \omega _1$ , $\beta < \lambda $ , and $\gamma = \lambda \cdot i + \beta $ , $g^l_{\gamma } \in (M^l_i)^\omega \cap V[G_{\alpha _i, \lambda _\gamma }]$ .

3. (3) For every $l < 2$ and $i < \omega _1$ , ${\langle g^l_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ .

4. (4) Each $\lambda _\gamma $ is an ordinal below $ \lambda $ . For $\lambda \cdot i \le \gamma \le \gamma ' < \lambda \cdot (i+1)$ , we have $\lambda _\gamma \le \lambda _{\gamma '}$ .

5. (5) For $i < \omega _1$ and $l < 2$ , $\{ g^l_\gamma : \gamma < \lambda \cdot i \} = \{ f^l_\gamma : \gamma < \lambda \cdot i \}$ .

6. (6) If $\lambda \cdot i \le \gamma < \lambda \cdot (i+1)$ , then $\mathcal {U}_\gamma \in V[G_{\alpha _i}, \lambda _\gamma ]$ .

7. (7) If $\gamma < \delta < \lambda \cdot \omega _1$ , then $\mathcal {U}_\gamma \subseteq \mathcal {U}_\delta $ .

8. (8) If $\gamma < \lambda \cdot \omega _1$ is a limit ordinal, then $\mathcal {U}_\gamma = \bigcup _{\delta < \gamma } \mathcal {U}_\delta $ .

9. (9) $X_\gamma \in \mathcal {U}_{\gamma + 1}$ or $\omega \smallsetminus X_\gamma \in \mathcal {U}_{\gamma +1}$ .

10. (10) If $\varphi (x_1, \dots , x_n)$ is an $\mathcal {L}$ -formula, $\gamma = \lambda \cdot i + \beta $ , and $\gamma _1, \dots , \gamma _n \le \gamma $ , then $Y_{\varphi , \gamma _1, \dots , \gamma _n}$ defined below belongs to $\mathcal {U}_{\gamma +1}$ :

    $$\begin{align*}&Y_{\varphi, \gamma_1, \dots, \gamma_n} = \{ k \in \omega : M^0_i \models \varphi(g^0_{\gamma_1}(k), \dots, g^0_{\gamma_n}(k))\\&\quad \Leftrightarrow M^1_i \models \varphi(g^1_{\gamma_1}(k), \dots, g^1_{\gamma_n}(k)) \}. \end{align*}$$

(Construction) First we let $U_0$ be the set of cofinite subsets of $\omega $ .

Suppose that ${\langle \mathcal {U}_\delta : \delta \le \gamma \rangle }$ and ${\langle g^0_\delta , g^1_\delta , \lambda _\delta : \delta < \gamma \rangle }$ are defined. Now we will define $g^0_\gamma , g^1_\gamma , \lambda _\gamma $ and $\mathcal {U}_{\gamma +1}$ . Take i and $\beta $ such that $\gamma = \lambda \cdot i + \beta $ .

Suppose that $\gamma $ is even.

Let $g^0_\gamma = f^0_{\varepsilon _\gamma }$ , where $\varepsilon _\gamma $ is the minimum ordinal such that $f^0_{\varepsilon _\gamma }$ does not belong to $\{g^0_\delta : \delta < \gamma \}$ .

Take $\lambda ' < \lambda $ such that $M^0_i, M^1_i, {\langle g^0_\delta : \delta \le \gamma \rangle }, {\langle g^1_\delta : \delta < \gamma \rangle } \in V[G_{\alpha _i, \lambda '}]$ . Put $\lambda _\gamma = \lambda ' + 1$ . Take a bijection $\pi ^1_i \colon \omega \to M^1_i$ in $V[G_{\alpha _i, \lambda '}]$ . Define $g^1_\gamma $ by $g^1_\gamma = \pi ^1_i \circ c^i_{\lambda '}$ .

Put $\mathcal {Y} = \{ Y_{\varphi , \gamma _1, \dots , \gamma _n} : \varphi (x_1, \dots , x_n) \text { is an } \mathcal {L}\text {-formula and }\gamma _1, \dots , \gamma _n \le \gamma \}$ . Now we show $\mathcal {U}_\gamma \cup \mathcal {Y}$ has the finite intersection property. In order to show it, let $X \in \mathcal {U}_\gamma $ , ${\langle \varphi _\iota : \iota \in I\rangle }$ is a finite sequence of $\mathcal {L}$ -formulas and $\gamma ^\iota _1, \dots , \gamma ^\iota _{n_\iota }$ for $\iota \in I$ are ordinals that are less than $\gamma $ . It suffices to show that the set $D \in V[G_{\alpha _i, \lambda '}]$ defined below is a dense subset of $\mathbb {C}$ :

$$ \begin{align*} &D = \{ p \in \mathbb{C}:\ (\exists k \in \operatorname{dom}(p) \cap X)(\forall \iota \in I) \\ & \qquad M^0_i \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\gamma(k)) \Leftrightarrow M^1_i \models \varphi_\iota(g^1_{\gamma^\iota_1}(k), \dots g^1_{\gamma^\iota_{n_\iota}}(k), \pi^1_i(p(k))) \}. \end{align*} $$

We now prove this. Let $p \in \mathbb {C}$ .

For each $k \in \omega $ and $\iota \in I$ , put

$$\begin{align*}v(k, \iota) = \begin{cases} 1, & \text{if }M^0_i \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots, g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\gamma(k)), \\ 0, & \text{otherwise.} \end{cases} \end{align*}$$

And for each $k \in \omega $ put

$$\begin{align*}v(k) = {\langle v(k, \iota) : \iota \in I\rangle}. \end{align*}$$

Then by finiteness of ${}^I 2$ , for some $v_0 \in {}^I 2$ , we have $\omega \smallsetminus v^{-1}(v_0) \not \in \mathcal {U}_\gamma $ .

For each $\iota \in I$ , put

$$\begin{align*}\varphi_\iota^+(x^\iota_1, \dots, x^\iota_{n_\iota}, y) \equiv \begin{cases} \varphi_\iota(x^\iota_1, \dots, x^\iota_{n_\iota}, y), & \text{if } v_0(\iota) = 1, \\ \neg \varphi_\iota(x^\iota_1, \dots, x^\iota_{n_\iota}, y), & \text{otherwise.} \end{cases} \end{align*}$$

Put

$$\begin{align*}\psi \equiv \exists y \bigwedge_{\iota \in I} \varphi_\iota^+(x^\iota_1, \dots, x^\iota_{n_\iota}, y). \end{align*}$$

Then by the induction hypothesis (10), $Y_{\psi , {\langle \gamma ^\iota _1, \dots \gamma ^\iota _{n_\iota } : \iota \in I\rangle }} \in \mathcal {U}_\gamma $ . So take $k \in X \cap v^{-1}(v_0) \cap Y_{\psi , {\langle \gamma ^\iota _1, \dots \gamma ^\iota _{n_\iota } : \iota \in I\rangle }} \smallsetminus \operatorname {dom}(p)$ .

Since $M^0_i \models \psi ({\langle g^0_{\gamma ^\iota _1}(k), \dots g^0_{\gamma ^\iota _{n_\iota }}(k) : \iota \in I\rangle })$ , we have $M^1_i \models \psi (\langle g^1_{\gamma ^\iota _1}(k), \dots g^1_{\gamma ^\iota _{n_\iota }}(k) : \iota \in I\rangle )$ .

By the definition of $\psi $ , we can take $y \in M^1_i$ such that $M^1_i \models \varphi _\iota ^+(g^1_{\gamma ^\iota _1}(k), \dots , g^1_{\gamma ^\iota _{n_\iota }}(k), y)$ for every $\iota \in I$ . We now put $q = p \cup \{ (k, (\pi ^1_i)^{-1}(y)) \} \in \mathbb {C}$ . This witnesses denseness of D.

Now we define $\mathcal {U}_{\gamma +1}$ as the filter generated by $\mathcal {U}_\gamma \cup \mathcal {Y} \cup \{ X_\gamma \}$ or the filter generated by $\mathcal {U}_\gamma \cup \mathcal {Y} \cup \{ \omega \smallsetminus X_\gamma \}$ .

When $\gamma $ is odd, do the same construction above except for swapping 0 and 1. Since the above construction below $\lambda \cdot (i+1)$ can be performed in $V[G_{\alpha _i + 1}]$ , (3) in the induction hypothesis holds. (End of Construction.)

Now we put $\mathcal {U} = \bigcup _{\gamma < \lambda \cdot \omega _1} \mathcal {U}_\gamma $ , which is an ultrafilter over $\omega $ . Then the function

$$\begin{align*}{\langle([g^0_\gamma]_{\mathcal{U}}, [g^1_\gamma]_{\mathcal{U}}) : \gamma < \lambda \cdot \omega_1\rangle} \end{align*}$$

witnesses $(M^0)^\omega / \mathcal {U} \simeq (M^1)^\omega / \mathcal {U}$ .

Proof Let $\mathbb {A}$ denote the amoeba forcing. Let $\lambda> \aleph _1$ be a regular cardinal with $\lambda ^{<\lambda } = \lambda $ . Let ${\langle \mathbb {P}_\alpha , \dot {\mathbb {Q}}_\alpha : \alpha < \omega _1\rangle }$ be a finite support forcing iteration such that for all even $\alpha < \omega _1$ we have $\Vdash _\alpha \dot {\mathbb {Q}}_\alpha = \mathbb {C}_\lambda $ and for all odd $\alpha < \omega _1$ we have $\Vdash _\alpha \dot {\mathbb {Q}}_\alpha = \mathbb {A}$ .

Then $\mathbb {P}_{\omega _1} \Vdash \operatorname {KT}(\aleph _1)$ by Theorem 5.1.

Moreover, we have $\operatorname {cof}(\mathsf {null}) = \aleph _1$ since the amoeba forcing $\mathbb {A}$ adds a null set containing all null sets coded in the ground model (see [Reference Bartoszyński and Judah2, p. 106]).

Question 6.1.

1. (1) Does $\operatorname {KT}(\aleph _1)$ imply a stronger hypothesis than $\mathfrak {mcf} = \aleph _1$ ? In particular does $\operatorname {KT}(\aleph _1)$ imply $\operatorname {non}(\mathsf {meager}) = \aleph _1$ ?

2. (2) Does $\operatorname {KT}(\aleph _0)$ imply a stronger hypothesis than $\mathfrak {c}^\exists \le \mathfrak {d}$ ? In particular does $\operatorname {KT}(\aleph _0)$ imply $\operatorname {non}(\mathsf {meager}) \le \operatorname {cov}(\mathsf {meager})$ ?

3. (3) In the Sacks model, does $\operatorname {KT}(\aleph _0)$ hold? (If in this model $\neg \operatorname {KT}(\aleph _0)$ holds, we can separate $\operatorname {KT}(\aleph _0)$ and $\mathfrak {c}^\exists \le \mathfrak {d}$ .)